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15 formulas that connect twin prime pairs
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15 formulas that connect twin prime
pairs
--------------------------------Odd number theory
----------------------------------------------WOLF Marc, WOLF François
10/01/2017
Abstract:
Odd numbers are written Np=2p+1 with
. One considers these numbers as the terms of
an arithmetic progression with a common difference equal to 2. A theory based on indices p
with
allows describing structure of odd numbers. This distribution of numbers allows
demonstrating the existence of 15 formulas that connect twin prime pairs except for pairs (3,
5) and (5, 7). The reciprocal is not true.
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15 formulas that connect twin prime pairs
Introduction .............................................................................................................................. 4
I- Set W and odd primes .......................................................................................................... 4
I.1 Preliminaries...................................................................................................................... 4
I.2 Set W ................................................................................................................................. 5
I.3 Subset of W ....................................................................................................................... 6
1.3.1 By fixing the variable j .............................................................................................. 6
1.3.2 Definition of elements of the set W ........................................................................... 8
1.3.3 By fixing the variable n.............................................................................................. 8
I.4 Remarkable number of the set
, basic interval and base unit associated with j ......... 10
II- Trigonometric functions associated with the set W ....................................................... 12
II.1 The set W ....................................................................................................................... 12
II.2 Trigonometric functions associated with a subset
................................................... 12
III- Sequence – arithmetic progression ................................................................................ 14
III.1 Definition of a sequence j of the set Wj ........................................................................ 14
III.2 Combination between n sequences Wj ......................................................................... 14
IV- The 3 parameters that define the number of pairs of twin prime ............................... 19
IV.1 Definition of the 3 parameters ...................................................................................... 19
IV.2 Relation with primes and pairs of twin primes............................................................. 20
V- Determination of the 3 parameters .................................................................................. 21
V.1 Determination of parameter A1 ..................................................................................... 21
V.2 Determination of parameter B1 ..................................................................................... 24
V.2.1 Enumeration and computing the value of B1 .......................................................... 24
V.2.2 B1(j) is a strictly increasing function ...................................................................... 29
V.2.3 Evolution of B1(j) to infinity .................................................................................. 31
V.2.4 Infinity of primes .................................................................................................... 38
V.3 Determination of parameter C1 ..................................................................................... 39
V.3.1 Relationship to count pairs of composite numbers ................................................. 39
V.3.2 Count pairs of composite numbers.......................................................................... 40
V.3.3 Enumeration and computing value of C1 ............................................................... 51
V.3.4 C1(j) is a strictly increasing function ...................................................................... 59
V.3.5 Evolution of C1(j) to infinity .................................................................................. 60
VI- The infinity of pairs of twin primes ............................................................................... 64
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15 formulas that connect twin prime pairs
VI.1 Enumeration of pairs of twin primes ............................................................................ 64
VI.2 New conjecture on twin primes .................................................................................... 66
VII- Formulas connecting the twin prime pairs .................................................................. 66
VII.1 Combination of 3 sequences W0, W1 and W2 ............................................................ 66
VII.2 Calculation of the number of pairs of virtual twin primes .......................................... 67
VII.3 Theorem: 15 formulas connecting the pairs of twin primes ....................................... 68
VIII- Infinity of pairs of virtual twin prime numbers ........................................................ 70
VIII-1- Definitions ................................................................................................................ 70
VIII-2- Determination of the number of pairs of virtual twin numbers in a natural period . 71
Conclusion ............................................................................................................................... 75
References: .............................................................................................................................. 76
Appendix 1 .............................................................................................................................. 77
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15 formulas that connect twin prime pairs
Introduction
Odd numbers are written Np=2p+1 with
. The p number is named the index of the Np
number. In this article, we present a new theory on odd numbers based on indices. The
distribution of indices of odd composite numbers is represented by the set W while the set M
is the set of indices of primes.
In the first three parts, the new theory is presented: the set W and its properties. In Part 4, we
demonstrate that the computation of the number of twin prime pairs is done using three
parameters:
- A1(p): number of pairs of odd numbers that are written in the form (6m-1, 6m+1)
A1(p)=card(S)
- B1(p): number of odd composite numbers that are written in 6m-1 or 6m+1
B1(p)=card(O)
- C1(p): number of pairs of odd composite numbers that are written in the form (6m-1, 6m+1).
C1(p)=card(T)
The formulas of these three parameters are demonstrated in the part 5. In this part, we
identified two specific alternating series which converge to 1.
In part 6, a new conjecture on the pairs of twin primes is presented.
Finally, we demonstrate, in the last part and using three parameters that there are 15 formulas
that connect pairs of twin primes except the first two pairs (3, 5) and (5, 7). A new theorem is
determined.
I- Set W and odd primes
I.1 Preliminaries
Definition 1.1:
Introduction of some notations:
1. Let I be the set of odd integers greater than 1, i.e.:
2. Let P be the set of prime numbers. This set is infinite [1].
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15 formulas that connect twin prime pairs
3. Let C be the set of odd composite integers greater 1, i.e.:
Thus
will be called index of number
.
4. This theory only concerns integer numbers. We will do the abuse of notation consisting
talking interval although this is not interval of real numbers but set of consecutive integers.
Example: [1, 3] = {1, 2, 3} et [1,3[ = {1, 2}. We will show its utility from the property 7.
I.2 Set W
Definition 1.2.1:
W is the sub-set of natural integers which are written:
avec
Definition 1.2.2:
The complement of W in N is denoted: M
The two increasing sequences associated with M and W are referenced in the On-Line
Encyclopedia Of Integer Sequences with the names A153238 and A067076.
Property 1:
Let p be a natural integer.
Thus in noting P the set of prime numbers, one gets:
Demonstration:
If
then there is a natural integer pair
the index of a composite number. Hence:
with
such as
is
But
and
with
are two integers greater than 2, so
is odd
composite number.
Reciprocally, if
is odd composite number then
. An odd composite
number decomposes uniquely into odd primes.
The multiplication of two odd numbers always produces an odd number. So an odd
composite number is divided into two odd numbers.
The smallest odd prime number is 3. This is the smallest factor of an odd number.
Each factor of an odd number can be represented by the following formula:
with
or
with
.
Let
be a composite number, this number can be factorized by
and
.
Hence
Or
The property is demonstrated.
In taking the contrapositive of implications of property 1, one gets property 2.
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15 formulas that connect twin prime pairs
Property 2
Let p be a natural integer:
written:
Property 2bis
Let p be a natural integer:
Thus in noting P as the set of prime numbers, one gets:
that can also be
.
Remark:
Knowledge of the set M immediately lets you know if an Np number is prime.
Knowledge of the set W immediately lets you know if an Np number is a composite
number. Knowledge of pairs of j and n integers gives a factorization of Np.
Example:
The first elements of the set M give first odd primes.
0
1
2
4
5
7
8
10
13
14
3
5
7
11
13
17
19
Definition 1.2.3 : Let p be a natural integer, p is called the index of the
23
29
31
number such as:
I.3 Subset of W
The definition 1.2.1 of W shows that by fixing one of the variables n or j, and varying the
other, one defines a subset of W. The set W is written:
1.3.1 By fixing the variable j
Elements of the set W are written as a function of n:
Definition 1.3.1:
With
,
. We denote
a sequence « j »
Property:
A sequence « j » is the set of indices of odd numbers which are multiples of the odd
number
.
Demonstration:
With
,
The property 1 says that if
.
, one gets:
So if « j » is taken as a constant, values of n allows to obtain the odd numbers that are
multiples of the odd number
.
The property is demonstrated.
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15 formulas that connect twin prime pairs
Consequence:
and
Remark:
By attributing small values to j and n, one determines the first terms of the sets W and M.
W = {3 ; 6 ; 9 ; 11 ; 12 ; 15 ; 16 ; …}
M = {0 ; 1 ; 2 ; 4 ; 5 ; 7 ; 8 ; 10 ; 13 ; 14 ; …}
Illustration 1:
The sets
,
,
,
can be represented in a coordinate system.

is represented by points which have an integer ordinate not equal to 0. These
points are on the following line:
=3n

is represented by points which have an integer ordinate not equal to 0. These
points are on the following line:
=5n+1

is represented by points which have an integer ordinate not equal to 0. These
points are on the following line:
=7n+2

is represented by points which have an integer ordinate not equal to 0. These
points are on the following line:
=9n+3
The crosses on the ordinate axis are k indices and so elements of the set W.
n=1
Ff
.
n=1
Ff
.
Ff
.
Ff
.
Ff
.
n=1
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15 formulas that connect twin prime pairs
1.3.2 Definition of elements of the set W
A « point » is shown on the graph by a pair of value (n, j).
An index is obtained by taking the ordinate of a point that is to say by the relationship k (n, j).
Each element of the set W is associated with one index or one to several points.
Definitions 1.3.2:
1. A « point » is defined by a pair of value (n, j).
2. An index is a numerical value obtained by the relation
. Each
"point" is associated with one index. Each index is associated with one or several points.
3. A subset Wj consists of a sequence of indices associated with a set of points. Each element
of the set Wj is associated with one index or one to several points.
4. The set W consists of sequences of indices associated with sets of points. Each element of
the set W is assigned to one index or one to several points.
1.3.3 By fixing the variable n
Elements of the set W are written as a function of j:
Definition 1.3.3.1:
Consequence:
With
,
and
Illustration 2: The sets
,
,
,
can be represented in a coordinate system.

is represented by points which have an integer ordinate. These points are on the
following line:
= 3j+3

is represented by points which have an integer ordinate. These points are on the
following line:
= 5j+6

is represented by points which have an integer ordinate. These points are on the
following line:
= 7j+9

is represented by points which have an integer ordinate. These points are on the
following line:
= 9j+12
The crosses on the ordinate axis are k indices and so elements of the set W.
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15 formulas that connect twin prime pairs
n=1
Ff
.
n=1
Ff
.
Ff
.
Ff
.
Ff
.
n=1
n=2
n=3
Property 3:
Demonstration:
Let p be an element of the set
Let
, with
and
So p is an element of the set
Let p be an element of the set
Let
, then
So p is an element of the set
Equality of the sets
. Then a non-zero natural integer n exists such as
. Then a natural integer n exists such as
and
.
is demonstrated.
Subsequently it is chosen to fix j.
Definition 1.3.3.2: With
and
The common difference of the set
is the difference between two consecutive indices of the
sequence "j". It is equal to
. Each element of the set Wj is a term of an
arithmetic progression u such as u0 = j and un+1 = un + (2j + 3).
Property 4: The common difference of the set
is equal to
Remark:
1. We called "period" the common difference of the arithmetic progression associated with
2. In the previous illustration, the period of the set
is the difference ordinate between 2
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15 formulas that connect twin prime pairs
consecutive points of the line
Property 5: If the period
of the set
already exist in at least one of the sets
Demonstration: With
such as
with
and
Let
be a composite number such as
with
,
One gets:
with
Let
and
is a composite number then elements of the set
with
. In other words:
,
and
be divisible by the element
a natural integer such as
With
and
, one gets:
Hence:
I.4 Remarkable number of the set
associated with j
, basic interval and base unit
Definition 1.4.1: With
,
The index with the value
of the set
with
the set . In the set W, this index is written
1.
value of j. It will be noted
is called a remarkable number of
. It is only dependent of the
Property 6: a remarkable number is written in the set W as following:
The odd number denoted
square of an odd number:
, associated to the remarkable number
.
is equal to the
Demonstration:
And
1
The index called GW is a tribute to our mother Géneviève WOLF
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15 formulas that connect twin prime pairs
The odd number
is the square of an odd number.
Definition 1.4.2: The value of j associated to the set
is the index of the common difference
of the arithmetic progression (Sequence "j") that is denoted . The value of the common
difference is equal to the square root of the odd number
:
Definition 1.4.3: With
With
, one defines
by the following formula:
Property 7:
Number
is a prime if and only if for
, one gets
Demonstration: See the property 2
Definitions 1.4.4:
1. With
, a base interval associated to j is defined as follows:
2. We call a base distance
3. The base unit
numbers. One denotes
of the set Wj the length of the base interval :
is defined as being the distance between two consecutive remarkable
=card( ) and so
.
4. The union of all the basic intervals forms a subset of N that is written
:
Examples:
j
0
1
2
3
4
5
6
7
3
11
23
39
59
83
111
143
7
11
15
19
23
27
33
Property 8: remarkable numbers cannot be written in the following mathematical
form:
Demonstration:
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15 formulas that connect twin prime pairs
But the value
is not divisible by 3.
Hence
II- Trigonometric functions associated with the set W
II.1 The set W
The set W is written:
a subset of W is written:
Let
Each natural integer p is associated with an integer equal to 2p + 3 which is either an odd
composite number when p is an element of W or an odd prime:
II.2 Trigonometric functions associated with a subset
Definition 2.2.1: At any subset
as:
, a function
If
, then one gets:
Remark :
is defined on the non-zero real numbers such
is the set of solutions of the equation
The period of the function
is equal to
,
.
Examples:
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
For
one gets:

For
one gets:

For
one gets:
Property 11:
The function
becomes zero if and only if
Examples:
1. Here is the graphical representation of the function
which becomes zero for all
multiples of 3 that are x=3*n. The set
can be written as following:
2) Here is the graphical representation of the function
numbers written as following:
. The set
Demonstration: if
With
Reciprocally if
then there is a natural integer
then
which becomes zero for all
can be written as following:
such as
. Thus:
.
, then
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15 formulas that connect twin prime pairs
Substituting into the function
, we get:
When j varies from 0 to , the product of functions
becomes zero for each element of
the set W such as
. We can state the following property:
Property 12:
Demonstration: It follows immediately from the property 11.
Remark: When tends to infinity, we get a characterization of the set W that is to say all
elements of the set W. However to define each element in this way is not faster than using the
definition 1.2.1.
III- Sequence – arithmetic progression
III.1 Definition of a sequence j of the set Wj
Definition 3.1: When
,
We call a sequence « j » of the set
the set of indices of odd numbers that are
multiples of the odd number
.
A sequence « j » also called sequence
is an arithmetic progression with a common
difference and a first term j such as
.
III.2 Combination between n sequences Wj
We will define what a combination of n sequences Wj is.
We will take an example with two sequences W0 and W1 with the respective common
differences r0=3 et r1=5. The sequences W0 and W1 define the two following functions which
gives indices
:
We define a combination of two distinct sequences when we assign a criterion for selection of
indices between these two sequences named, for example, A and B, such as
. The combination is represented by the set of indices respecting the criterion.
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15 formulas that connect twin prime pairs
Elements of the set Wj are represented by an arithmetic progression and belong to . When
one applies to 2 arithmetic sequences A and B, a selection criterion such as
,
each element of the resulting set C is a pair of integer numbers and belongs to . The first
element of the pair is taken in the first sequence Wj1 and the second one in the second
sequence Wj2.
Examples of combinations of two sequences:
The 1st combination with the criterion: k1(y) = k0(x)
Indices matching this criterion are determined by solving a linear Diophantine equation:
.
The particular solution is the pair (x0, y0) . In the above example, the pair is (2,1). The set of
solutions is obtained with x=5n+2 and y =3n+1 that gives a pair of indices
,
. The first element of the pair is taken in the first sequence W0 and the
second one in the second sequence W1.
The indices
respecting the criterion are represented by an arithmetic progression with a
common difference equal to the product of common differences between W0 and W1 and a
first term equal to index of :
The 2nd combination with the criterion: k1(y) = k0(x) + 1. The linear Diophantine equation to
solve is:
The particular solution is the pair 0, 0). The set of solutions is obtained with x=5n and y =3n
that conducts to pairs of indices
.
The 3rd combination with the criterion: k1(y) = k0(x) + 2. The linear Diophantine equation to
solve is:
The particular solution is the pair (3, 2). The set of solutions is obtained with x=5n+3 and y
=3n+2 that gives a pair of indices
.
The indices of the pairs of indices of the three previous combinations are written as an
arithmetic sequence with the same common difference equal to
. The number of
distinct combinations is related to the number of elements of each set Wj existing in an
interval with a length equal to the product of common difference of each arithmetic sequence,
less 1:
. In this interval, there are elements of the set W0 consecutively
separated by a distance equal to and elements of the set W1 consecutively separated by a
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15 formulas that connect twin prime pairs
distance equal to . The maximum distance between an element of W0 and an element of W1
is equal to r-1. The number of distinct combinations is then equal to
.
A combination exists only once in each interval of length
consecutive. Combinations
respect a criterion of selection of indices between two sequences, i.e. two arithmetic
sequences A and B, such as
. A unique criterion of
selection exists for each distinct combination.
When one of the Wj1 and Wj2 sequences has a common difference that is a multiple of the
other, only one criterion leads to a solution. If
, the only existing
combination between the Wj1 and Wj2 sequences is obtained with the criterion: kj2(y) = kj1(x).
The set Wj2 is a subset of the set Wj1. With W0 and W3 sequences, r0=3 and r3=9, the only
existing combination is obtained with the criterion: k3(y) = k0(x). The equation to solve is:
9y+3=3x. The pairs of indices are (9n+3, 9n+3), that is to say, indices of the set W3.
The combinations between n sequences Wj, studied in this article, are those obtained with the
sequences Wj and
.
Definitions 3.2:
3.2.1) We call combination between n sequences Wj, n set of indices whose indices A and B
of two distinct sequences Wj1 and Wj2, taken among n, match a criterion of the form:
.
3.2.2) Let
be the common difference of a sequence
. The common difference is also called period. The product of n distinct
common differences is equal to
. call combination between n distinct
sequences Wj, n set of indices whose indices of two distinct sequences Wj1 and Wj2, taken
among n, that is to say, two arithmetic progressions A and B match a criterion as follows :
. The set of criteria used to form a combination is unique.
The number of distinct sets of criteria is related to the number of indices of each sequence
in the period . This number of indices for a sequence
is equal to
. An index
by sequence
is selected to form a unique combination. This combination is repeated for
each consecutive period . A distinct combination formed by n sequences
is constituted
by n sets of indices. The number of distinct sets of criteria, that is to say the number of distinct
combinations is equal to:
3.2.3) Let A and B be two arithmetic progressions that are written respectively
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15 formulas that connect twin prime pairs
The odd numbers P1 and P2 are prime, then there is always a solution for the previous linear
Diophantine equation:
.
:
The difference between two odd numbers is an even number. There is always a pair of values
(y0, z0) such as the above mathematical relation is true.
The combination of two sets Wj
and B’ wh ch
w
:
p
m h m c lly by h w
f
c
A’
The l
A’ and B’ are arithmetic progressions with a period equal to the product of
period P1 and P2 of the arithmetic progressions A and B:
. Values of A’ and B’
are respectively a subset of values of A and B.
p
f v lu (A’( ), B’( )) x
once in an interval of length equal to P1*P2. When
, pairs of values
(A’( ), B’( )) are unique in an interval of length equal to and are repeated for each
consecutive interval.
In an interval
, according to the previous definition 3.2.2, number of
combinations of 2 sequences Wj is equal to
. Only one solution
exists for each value
that is a number of distinct combination equal to
. When the first value p1 of the interval [p1, p1+r[ belongs to the interval [0, r[, the set
of distinct combinations is obtained..
Property 13:
At each sequence
exists a value of j. With
and
, n sequences
are
defined. The set of distinct combinations between n sequences is present in the interval of
length equal to twice the product
of the period of each sequence less one:
The number of indices in the previous interval is equal to:
One combination of n sequences
is constituted by n sets of indices. Each set of indices
is defined by an arithmetic progression with a period (Common difference) equal to :
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15 formulas that connect twin prime pairs
In an interval of length equal to , the existing combinations of n sequences are all distinct
from one another. Each set of indices of a combination is written as an arithmetic sequence
with a common difference equal to r. Each set of indices of a combination is represented by a
sinusoidal function with a period equal to . The symmetry property of a sinusoidal function
allows us to take into account a period only equal to .
The common difference of each set of indices of a combination of n sequences is equal to .
We will denote that the period
of a combination of n sequences is equal to .
Proof:
It follows from the above definitions 3.2 and properties of the sinusoidal function. Indeed,
each sequence can be represented by a sinusoidal function with a period equal to
. Sinusoidal functions have two fundamental properties:
-
one periodicity
one Symmetry
The symmetry property allows us to take into account only the half period of the sinusoidal
function as a useful period. Indeed, the combinations are the same in the second half of the
period. The period of a sequence is then equal to
.
It is the same with a combination between n sequences . Each set of indices of the
combination may be represented by a sinusoidal function with a period equal to
. The useful period is equal to
.
Property 14:
At each sequence
exists a value of
. When n consecutive sequences
set of n periods
and a period
with
,
are selected, from the first sequence
, the
is the set of n first primes [P0, Pn-1]. The set of combinations of n
consecutive sequences whose the first sequence is
product of the first n primes less one.
is in a period
equal to 2 times the
The period of each set of indices of a combination of n consecutive sequences, from the
sequence
, is equal to
. The
period
of a combination of n consecutive sequences, from the sequence
natural period:
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, is called
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15 formulas that connect twin prime pairs
Proof:
It follows from the 5 and 13 property.
If the value of the period
of a
sequence is a composite number, the value
decomposes into prime factors. The sequence
is a subset of each sequence
which has a
period equal to one of the prime factors such that
. The natural period is the smallest set
containing all combinations of n consecutive sequences from
If the value of the period
of a
sequence is a composite number, the value
decomposes into prime factors. The sequence
is a subset of each sequence
which has a
period equal to one of the prime factors such that
. The natural period is the smallest set
containing all combinations of n consecutive sequences from
.
IV- The 3 parameters that define the number of pairs of twin prime
Definition 4:
Let E’A be the set of pairs of natural integers in the form
.
Remarks:
1. It may be noted that
and so it is a pair of odd numbers.
With the exception of the first two primes 2 and 3, the prime numbers are written in the form
6m-1 or 6m+1.
These numbers are not multiples of 2 or 3.
2. With the exception of the pair of twin primes (3.5), the pairs of twin primes are written as
follows (6m-1; 6m+1).
IV.1 Definition of the 3 parameters
Definition of the first parameter A1:
The parameter A1 is the cardinality of the set E’A. Therefore:
A1 = card(E’A)
Proposal: The A1 parameter is the number of pairs of indices (3n+1; 3n+2) with n
natural integer.
Demonstration: Let p be a natural integer which is index of the
integer such as:
Cas 1 :
Cas 2 :
The number of pairs of values (6m-1; 6m+1) is equal to number of pairs of indices (3n+1;
3n+2). EA is the set of pairs of indices:
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15 formulas that connect twin prime pairs
A1 = card(EA) = card(E’A)
We denote E’2A the set of odd numbers which are written in the form 6m-1 or 6m+1. The
indices of these odd numbers are written in the form 3n+1 or 3n+2.
The number of these elements is denoted A2:
A2 = card(
) = card(
)
Definition of second parameter B1:
The parameter B1 is the number of composite numbers belonging to the following pairs of
values (6m-1; 6m+1). It is also the number of indices 3n+1 and 3n+2 of the previous
composite numbers. These composite numbers form the set EB which is a subset of the set
E2A.
With
B1=card(
)
Definition of third parameter C1:
The parameter C1 is the number of pairs of composite numbers in the form (6m-1; 6m+1). It
is also the number of pairs of indices (3n+1; 3n+2). These pairs form the set EC which is a
subset of the set EA.
With
C1=card(
)
IV.2 Relation with primes and pairs of twin primes
Definition of the number of primes :
The function
gives the amount of prime numbers that do not exceed x. The number of
prime numbers in the interval [5, Np] is equal to
because we have to
remove the primes 2 and 3.
The value
is the number of primes that are written in the forms 6m-1 and 6m+1 in the
interval [0, Np]. It is also the number of indices 3n+1 and 3n+2 which are the indices of
primes. The set of these primes ENP is the complementary set of EB in E2A.
Property 1-1: when the enumeration of indices of the odd numbers, 3n + 1 and 3n +
2, is performed between 0 and index of the square of an odd number, the number of pairs of
indices is equal to half the number of indices:
A2=2 A1
Demonstration: It is immediately deducted from the property 8. To recall, the index of the
square of an odd number is named remarkable number.
The number of prime numbers in the form 6m - 1 or 6m + 1 is equal to:
=2 A1 - B1
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Definition of the number of pairs of twin primes
:
The function
gives the amount of pairs of twin primes whose primes do not exceed x.
The number of pairs of twin primes in the interval [5, Np] is equal to
because we have to remove the pair of twin primes (3, 5).
The value
is the number of pairs of twin primes that are written in the form (6m-1,
6m+1) in the interval [0, Np]. It is also the number of pairs of indices (3n+1, 3n+2) with 3n+1
and 3n+2 that are the indices of the twin primes.
We denote ECM the set of pairs (6m-1; 6m+1) whose only one number is a composite
number. The number of these pairs is denoted M1:
M1=card(ECM)
With M the complementary set of W in (Definition 1.2.2 and property 2bis), one gets:
The set of these pairs of twin primes named ECNPJ is the complementary set of ECM and EC in
EA.
We get:
A1 =
+ M1 + C1
B1 = 0 + 1*M1 + 2*C1
B1 = 2*C1 + M1
B1- C1 = M1 + C1
A1 =
+ B1 - C1
The number of pairs of twin primes written as follows (6m-1; 6m+1) is
= A1 - B1 + C1
equal to:
V- Determination of the 3 parameters
We will enumerate the elements of the sets EA, EB and EC to determine the value of the
parameters A1, B1 and C1. This enumeration is done with the indices 3m + 1 and 3m + 2 of
the odd numbers which are written in 6m -1 and 6m + 1. We will demonstrate that 3
parameters are dependent on the j parameter as a function of the second degree.
V.1 Determination of parameter A1
Enumeration of pairs of indices (3m+1; 3m+2) is done in counting the number of times the
index p=3m+2 exists in the interval [0, Np]. The number of pairs is equal to:
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When p=2, there is one pair. Hence the shift of index by one:
The number of pairs A1 is an integer number, hence the use of the integer part:
Property 2-1: the number of existing pairs of indices (3m+1; 3m+2) in the interval [0, Np]
with Np an odd number written as Np=2*p+3 is obtained by the following relation:
A remarkable number is written:
The Enumeration is done for each sequence
up to remarkable number of the following
sequence
included. That is to say for a selected sequence , the Enumeration is done in
the interval
.
By substituting
in formula A1(p), we get:
A1(j) = [
]
And
A2(j) = [
]
The number of integer values in a base unit j is equal to base distance of
that is
. This number is greater than 3, whatever the value of j. The A1 value strictly increases with
the variable j.
Example: table of the first values of A1 and A2 with j
j
A1
A2
0
4
8
1
8
16
2
13
26
3
20
40
4
28
56
5
37
74
6
48
96
Illustration 3: the graph of equation A1 = f(j) (Trend curve) and the regression coefficient R
were obtained with the Maple tool. The data are in black color and the trend curve is in red
color. The calculation was made until j = 5 000 170. For j = 4 999 998, the odd number Np =
1014 +1 is reached.
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15 formulas that connect twin prime pairs
A1(j) = 0.666666666668230 j2 + 3.33333746138579 j + 4
R = 0.999999999999944
Property 2-2: A1(j) is a strictly increasing function:
Remark: when the value of j is equal to a multiple of the GCD(p+1;3), A1(j) is written in the
form of a second degree polynomial. Hence the following property with
and
:
Property 2-3:
Property 2-4: There is infinity of pairs of odd numbers that are written as (6m-1; 6m+1).
~
~
Remarks : For recall :
1. We say that and
are two equivalent functions in infinity if
And we then denote
2. We say that
is dominated by
~
in infinity if the quotient is bounded. Therefore:
And we then denote
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15 formulas that connect twin prime pairs
V.2 Determination of parameter B1
In the interval [0, Np] with Np an odd number, we will determine the number of existing
composite numbers: B1 =card(EB). To do this, we will enumerate the number of indices 3m +
1 and 3m + 2 whose the associated odd numbers, respectively 6m - 1 and 6m + 1 are
composite numbers that is to say belonging to the set W. This enumeration will allows us to
determine the evolution of B1 as a function of j.
V.2.1 Enumeration and computing the value of B1
The set W contains all odd numbers multiple of an odd number. This set consists of all
sequences
with
.
Thus for all natural integer j,
and
And
A sequence
is an arithmetic progression with a common difference
first term j. Prime numbers does not belong to the set W because
.
and a
When the period is a composite number, the sequence
is a subset of a previous
sequence. The value of the period of the previous sequence is the smallest prime number
which factorizes the sequence . The enumeration is done only with the sequences whose
the period is a prime number.
Example: the sequence
has a period equal to 25. Odd multiple of 25 represent a subset
of odd multiples of five. The sequence
has a period equal to 5.
Definition 5.2.1: The enumeration is done in the following set
. The sequence
is represented by multiples of the prime number .
Property 3-1:
The number of values multiples of
is obtained, with
, by the following relation
.
Demonstration:
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Example:
The sequence
is represented by multiples of the prime number 5. The number of values
multiples of 5 is obtained by the relation
.
Property 3-1 bis:
We denote for convenience:
.
.
In the interval [0 , Np] with Np an odd number, the existing composite numbers are multiples
of prime numbers that are present in the interval [0 ,
].The enumeration is performed
using prime numbers present in the preceding interval. The number of prime numbers is equal
to Nmax + 1. The first odd prime number is P0=3 and the last one is PNmax.
The number of composite numbers that are multiple of a prime number is obtained by the
following relation:
Some of the composite numbers are multiples of several primes including the integer 3.
The enumeration realized to get value of B1 is to count each composite number once. When a
composite number can be written as a multiple of several prime numbers, this composite
number is counted several times. We will call b2 the number of times that a composite
number is counted more than 1 time. The value of parameter B1 is then equal to b1-b2.
Enumeration for getting value of b2 in the interval [0, Np]:
A composite number multiple of 5 and 7 will be counted 2 times in the formula of B1. One
must then reduce the value of B1 by the number of times that the primes {5, 7} are factors of
a same composite number. We say that prime numbers are common to composite numbers.
However, the composite numbers multiples of 3 must not be counted. We must also removed
the number of times that the prime numbers {3, 5}, {3, 7} are common to composite numbers.
When the prime numbers {3, 5} and {3, 7} are common to the same composite number, we
must remove only once the presence of these primes. To do this, we must add the number of
times that the prime numbers {3, 5, 7} are common to the same composite numbers.
The number of times that two primes are common to a same composite number is to remove
from the enumeration. It is the same every time the number of prime numbers common to a
same composite number is an even number
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The number of times that three primes are common to a same composite number is to add
from the enumeration. It is the same every time the number of prime numbers common to a
same composite number is an odd number.
Definitions 5.2.2:
1) We call i the number of prime numbers common to a same composite number.
The addition (+) or removing (-) to the enumeration is determined by the following relation:
signF = (-1)i+1.
2) The indices of the composite numbers that are multiples of the 2 primes P1=2*j1+3 and
P2=2*j2+3 are represented by a subset Wj1j2 of the sequences Wj1 and Wj2 such as:
This subset Wj1j2 represents the common indices to the sequence Wj1 and Wj2.
Let NF(Ci=2)=card(
) with C2={P1,P2} and
.
With a number of primes equal to i, NF(Ci) is equal to the number of times that primes, Pk1 to
Pki, are common to a same composite number present in the interval [0, Np]:
with
with
and
The demonstration for NF(Ci) will be proposed after the numeric example below.
We need know the combinations of prime numbers in the set of prime numbers consisting of
Nmax + 1 elements.
The table below gives the number of combinations obtained with Nmax = 4 for combinations
composed of 1 to 5 prime number that is from i = 1 to i = 5.
Compute
i
SignF=(-1)i+1
Number of
combinations of
primes
Comment
General case
for Nmax
for Nmax=4
b1
1
+
P0 = 3 is excluded
Search 1 element
among 4
4
[P1 ; P4]
b2
2
-
10
Search 2 elements
among Nmax+1=5
b2
3
+
10
Search 3 elements
among Nmax+1=5
b2
4
-
5
Search 4 elements
among Nmax+1=5
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15 formulas that connect twin prime pairs
b2
5
+
1
The combinations of prime numbers with Nmax = 4 are given in the table below. The
combinations of every prime number with prime numbers less than it are also given.
i
SignF=(-1)i+1
P1=5
P2=7
P3=11
P4=13
b1
1
+
(5)
(7)
(11)
(13)
b2
2
-
(3*5)
(3*7)
(3*11)
(3*13)
(5*7)
(5*11)
(5*13)
(7*11)
(7*13)
(11*13)
b2
3
+
(3*5*7)
(3*5*11)
(3*7*11)
(5*7*11)
(3*5*13)
(3*7*13)
(3*11*13)
(5*7*13)
(5*11*13)
(7*11*13)
b2
4
-
(3*5*7*11)
(3*5*7*13)
(3*5*11*13)
(3*7*11*13)
(5*7*11*13)
b2
5
+
(3*5*7*11*13)
Here is the calculation of NF(Ci) with i=2 and P1=3, P2=5:
The indices of integers multiples of 3 are p=3*x. They are the subset W0.
The indices of integers multiples of 5 are p=1+5*y. They are the subset W1.
The number of times that the two primes 3 and 5 are common to a same composite number,
by example Np=15=3*5, is determined when value of index p is the same in both subset.
That is to say when
.
A linear Diophantine equation must be solved. The solutions are:
Hence
When p=6, NFi=2({3,5}) is equal to 1. There is a shift of n by one. Thus:
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15 formulas that connect twin prime pairs
Example:
Np
p
n
15
6
1
0
45
21
2
1
75
36
3
2
We can write the following relation of b2:
with
with
and
Demonstration for NF(Ci):
We will determine the number of times that two prime numbers P1 and P2 are common to a
same composite number. That is to say find the number of composite numbers which can be
factorized by the prime factors P1 and P2. For this, the indices obtained with P1 and P2 must
be identical.
Indices of integers multiple of P1 are written:
Indices of integers multiple of P2 are written:
Hence
We must solve a linear Diophantine equation. The solutions are:
So
Proof:
Just replace x and y in the Diophantine equation.
The indices of composite numbers having as prime factors P1 and P2 are represented by an
arithmetic progression with a period (Common difference) equal to the product of primes and
a first term equal to the index of the product of primes. This relationship with the 2 primes is
similar to that obtained with one prime number.
With 3 prime numbers, a linear Diophantine equation must be solved such as:
With Nmax prime number, a linear Diophantine equation must be solved such as:
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15 formulas that connect twin prime pairs
The solutions are:
Let PF(Ci) be the product of prime numbers to the number of i:
The number of composite numbers which decompose with primes to the number of i is equal
to the factor n + 1. The shift of n by 1 is necessary because when p is equal to the first term of
the series, n is equal to 0 but it must be equal to 1.
We get the relationship:
Thus the formula of b2 is:
To recall, an odd number is written: Np=2*p+3 with
. Hence the property 3-2:
Property 3-2:
The number of existing composite numbers B1 in the interval [0, Np] is determined from the
prime numbers present in the interval
. Number of primes is equal to Nmax+1.
With the variable p, one gets:
V.2.2 B1(j) is a strictly increasing function
Index of an odd number Np is equal to p. The enumeration for B1 is conducted between 0 and
the square of an odd number. The index of the square of an odd number is called remarkable
number. This number is written depending on j such that:
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15 formulas that connect twin prime pairs
The enumeration is performed for each sequence
up to remarkable number of the next
sequence
. In other word, for a selected sequence , enumeration is performed in the
interval
.
Function B1(p) is written according to the parameter j if we replace p by
:
With
We will demonstrate that the number of composite numbers B1 is strictly increasing with j.
That is to say:
. B1(j) is a cumulative function. When value of j
increases, Np also increases, hence B1(j+1) ≥B1(j). When adding to B1 some composite
numbers that are multiple of a selected prime number, only those non common to previous
ones are counted. The B1 value can therefore only grow with the prime numbers
.
We will calculate the number of composite numbers multiple of the prime number P1=5
whose index of composite number is written as 3m+1 or 3m+2. The indices of composite
numbers are shifted by 1 compared to the indices of composite numbers multiple of 3.
The indices of the composite numbers that are multiples of 3 are: p=3x
The indices of the composite numbers that are multiples of 5 are: p’=5y+1
The indices of the composite numbers that are multiples of 3 and written as 3m+1 are:
p+1=5y+1 thus p=5y
We must solve the following linear equation Diophantine:
3x=5y thus 3x-5y=0
The solutions are y=3m and x=5m, that is to say p=15m+1.
The indices of the composite numbers that are multiples of 5 and written as 3m+2 are:
p+2=5y+1 thus p=5y-1
We must solve the following linear equation Diophantine:
3x=5y-1 thus 5y-3x=1
The solutions are y=3m+2, that is to say p=15m+11.
Every 15 consecutive indexes from index 1, there is a composite number multiple of 5 with an
index written as p=3m+1.
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15 formulas that connect twin prime pairs
Every 15 consecutive indexes from index 11, there is a composite number multiple of 5 with
an index written as p=3m+2.
Performing the calculation with the formula B1:
There are 2 prime numbers: P0=3 and P1=5
Hence Nmax=1
And so
With p=1+15*n, we get: B1 = 3*n - n = 2*n
There are 2 composite numbers multiple of 5 and non multiple of 3 every 15 consecutive
indices from p=1.
The number of indices in a base unit is equal to 4j+7. When j is greater than 2 the number of
indices is greater than 15. For j=0, 1 and 2, numerically, there is at least one composite
number multiple of 5 and not multiple of 3. Hence the property 3-3:
Property 3-3:
Number of composite numbers B1(j) is a strictly increasing function:
V.2.3 Evolution of B1(j) to infinity
We will demonstrate that B1(j) evolves as a second degree polynomial function.
In the interval [0, Np], the number of existing composite numbers B1 is determined from the
prime numbers present in the interval
. Number of primes is equal to Nmax+1.
A remarkable number is written:
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15 formulas that connect twin prime pairs
The enumeration is performed for each sequence
up to remarkable number of the next
sequence
. In other word, for a selected sequence , enumeration is performed in the
interval
.
Let
, we get the following formula of B1(j):
When j is equal to a multiple of the lowest common multiple (LCM) between all PF( ), B1(j)
is written as a second degree polynomial function. The LCM(PF( )) is equal to the product
of all prime numbers
.
We will determine value of both integer parts when Nmax tends to infinity:
Property 3-4a: When number of prime numbers Nmax tends to infinity, the sum of
with n belonging to interval [1, Nmax] is equal to:
Proof:
For each prime number greater than 23, the value of
is equal to -1. There are 7 prime
numbers in the interval [5;23]. Hence the result for n>7 is equal to 7The sum is then equal to 3 + 7 - = 10 – .
.
Property 3-4b: When number of prime numbers Nmax tends to infinity, the value of the
second integer part with n belonging to interval [0, Nmax] is equal to -2:
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15 formulas that connect twin prime pairs
Proof:
So the sum is equal to
.
Property 3-4: Let the following relation:
The value aB is a special alternating series that converges to 2/3 when the value of Nmax
tends to infinity that is to say when j tends to infinity.
The following special alternating series converges to 1:
Demonstration:
We can then write in using the precedent properties:
But B1=A2 And
(Property 2-3)
Hence
When j tends to infinity, the number of prime numbers Nmax tends also to infinity. We can
then also write:
We will write aB as a series. The table below shows the factorization of the series by the factor
1-1/P0.
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15 formulas that connect twin prime pairs
i
Sign
Decomposition of aB
Decomposition with factor 1/P0 and P0=3
SignF
1
+
2
-
3
+
4
-
x
(-1)x+1
+
+
+
+
The addition of terms of the table taking into account the sign leads to factorize aB with the
factor
We can then write:
Moreover, the first term b1 can be written as the second term b2:
Thus, we get:
The value of aB is written as an alternating series linked to the alternation of the sign (-1)(i+1).
But this is a special series because of the term of the series which depends on “ ” and Nmax.
It is a double sum. The term is written as follows:
The demonstration is done.
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15 formulas that connect twin prime pairs
Remark: In the table below, values of aB are given according to the number of primes Nmax
up to the first 60 prime numbers. Values are calculated for the first 4 and 5 elements of the
series that are
Nmax used in
Nmax
1
,
with
.
2
4
6
8
12
16
20
25
30
40
60
coeff aB
(4, Nmax)
0,2 0,209523 0,282983 0,324159 0,349656 0,380369 0,398644 0,410862 0,421316 0,428798 0,438233 0,447714
coeff aB
(5, Nmax)
0,2 0,209523 0,283050 0,324633 0,350834 0,383469 0,403994 0,418574 0,431956 0,442345 0,457103
Illustration 4:
0,5
0,45
0,4
0,35
coeff aB (4)
0,3
coeff aB (5)
0,25
0,2
0,15
0
10
20
30
40
50
60
70
Nmax
The limits at infinity of B1:
The prime number theorem [3] was proved independently by Jacques Hadamard and Charles
Jean de la Vallée-Poussin in 1896. In number theory, the prime number theorem (PNT)
describes the asymptotic distribution of the prime numbers among the positive integers. The
number π(x) f p m l
h
qu l x
qu v l wh h
lx
+∞ x
divided by its natural logarithm:
This number includes the primes 2 and 3 that are not in form 6m-1 or 6m+1. We can then
write:
.
We know that
=2 A1 - B1 (See IV.2 Relation with primes and pairs of twin primes)
And 2A1(j) =
With x = 2p+3 and p =
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15 formulas that connect twin prime pairs
When j tends to infinity, we can write:
By replacing to the numerator Pn by 2j+3, in formula B1 (1) available in beginning of this
paragraph (V.2.3 Evolution of B1(j) to infinity), we get:
Number of composite numbers is then written:
Hence:
with
We know that
The constant of Meissel–Mertens [2] is written:
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15 formulas that connect twin prime pairs
With x = 2p+3 and p =
, we get:
Hence after eliminating the constant:
Hence the property presented below.
Property 3-5:
There are endless numbers of composite numbers in form 6m - 1 or 6m + 1. The limit to
infinity of B1(j) can be written:
Or
The number of composite numbers evolves at infinity according to a second degree
polynomial function.
Illustration 5: the graph of equation B1 = f(j) (Trend curve) and the regression coefficient R
were obtained with the Maple tool. The data are in black color and the trend curve is in red
color. The calculation was made until j = 5 000 170. For j = 4 999 998, the odd number Np =
1014 +1 is reached.
B1(j)= 1.20999899917857 j2 - 26204.7633708626 j + 1
R = 0.999999899713826
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15 formulas that connect twin prime pairs
V.2.4 Infinity of primes
The study of pairs of twin primes is performed on the knowledge of the infinity of primes. We
will write this infinity depending on the parameter j.
We know that the number of primes
is equal to 2*A1 - B1=A2-B1.
= A2 - B1
When j = LCM(3,
)*n and n tends to infinity, we can write:
Or
The number of primes tends to infinity as a function related to j.
Note:
is an increasing function. Legendre's conjecture indicates that this function is
strictly increasing. The following illustration with the numerical values obtained show that
this is true at least until 1014.
Remark: the number of primes obtained in the interval [0, 10t] with
is an integer
sequence referenced in the On-Line Encyclopedia of Integer Sequences (OEIS) as A006880.
Our calculation is performed only with numbers in form 6m-1 or 6m + 1. The values 2 and 3
are excluded. So we get the same values less 2.
Illustration 6: the graph of equation
(Trend curve) and the regression coefficient R
were obtained with the Maple tool. The data are in black color and the trend curve is in red
color. The calculation was made until j = 5 000 170. For j = 4 999 998, the odd number Np =
1014 +1 is reached.
= 0.123334334076949 j2 + 26211.4303085093 j + 7
R = 0.999991214193368
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15 formulas that connect twin prime pairs
V.3 Determination of parameter C1
In the interval [0, Np] with Np an odd number, we will determine the number of existing pairs
of composite numbers: C1 =card(EC). To do this, we will enumerate the number of pairs of
indices (3m + 1, 3m + 2) whose the associated odd numbers, respectively (6m – 1, 6m + 1)
are composite numbers that is to say belonging to the set W. This enumeration will allows us
to determine the evolution of C1 as a function of j. This enumeration is performed with
primes in the interval
. The number of primes is equal to Nmax+1.
As B1, enumeration is performed in the following set:
Recall: Let j be a natural integer and
We call period (Common difference) of the set
of .
, the gap between two consecutive values
V.3.1 Relationship to count pairs of composite numbers
We will define a method to count the number of pairs of composite numbers (6q-1, 6q + 1)
and so the number of pairs of indices (3m + 1, 3m + 2) present in interval [0, Np].
Definition 5.3.1:
The pairs of composite numbers (6q-1, 6q + 1) are multiple of pairs of prime numbers (P1,
P2). In a such pair (6q-1, 6q + 1), we will write that:
The first composite number is multiple of a prime number P1
The second composite number is multiple of a prime number P2
Each composite number of a pair of composite numbers is multiple of a prime number of a
pair of primes (P1, P2). Each prime number Pj correspond to a period
of a set .
Indices (3m+1, 3m+2) of composite numbers are written depending on primes as follows:
For a pair of primes (P1, P2), in an interval [0, Np], number of pairs of composite numbers
associated to the pair (P1, P2) is obtained with indices p2 such as p2=p1+1. These pairs
represent the set EC(j1,j2) which is a subset of the set E C.
With C1j1,j2=card(
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15 formulas that connect twin prime pairs
We will look for pairs of composite numbers that are not multiples of 3 and whose indices are
separated by one unit that is to say with the indices (3m + 1, 3m + 2). For this, we will
determine all pairs of composite numbers multiple of pairs of primes (P1, P2) and (P2, P1).
V.3.2 Count pairs of composite numbers
Property 4-1:
The 2 composite numbers of a pair whose indices are (3m+1, 3m+2) cannot be multiple of a
same prime number.
Proof:
Gap between indices of 2 composite numbers of a pair (3m+1, 3m+2) is equal to 1.
Minimum gap between indices of 2 consecutive composite numbers belonging to a sequence
is equal to value of the period . The lowest value is equal to the smallest prime number
greater than 3. That is to say for j=1 with
Indices of composite numbers multiples of 5 are written: p=5*m+1.
Minimum gap between indices of 2 consecutive composite numbers of a set
is then equal
to 5.
Definition 5.3.2:
The criteria to find pairs of composite numbers (X, Y) are:
- 2 composite numbers written respectively as (6q-1, 6q+1) with Y-X=2
or
- 2 indices (p1, p2) of composite numbers whose indices are written respectively (3m+1,
3m+2) and p2 – p1 = 1.
Property 4-2:
Let (p1, p2) be indices of a pair of composite numbers multiple respectively of each prime
number of a pair of prime numbers (P1, P2) such that P1 and P2 are written 6q-1 and/or 6q +
1. A method to determine indices (p1, p2) is to solve successively 2 linear Diophantine
equations. There is always a solution because the coefficients of the equation are primes. The
solutions are arithmetic sequences. The indices p are written as p = a + b * n. Indices of pairs
are (p-1, p) with "a" corresponding to the period. Values of the coefficients "a" and "b"
depend on P1 and P2.
Proof:
a) First linear Diophantine equation
Indices of composite numbers multiple of P1 are written:
Indices of composite numbers multiple of P2 are written:
Indices are separated of one unit. A pair (p1, p2) can be written (p, p+1).
We search indices (p, p+1) related to a pair or primes (P1, P2) with p=p1 such as:
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15 formulas that connect twin prime pairs
Hence
We must solve a linear Diophantine equation.
Solutions exist because P1 and P2 are primes.
The solutions are written in the form a*n + t. It is an arithmetic progression.
By replacing x in the formula (1), one gets:
With
By replacing y in the formula (2):
with
The value of "a" is equal to the product of the primes. The t0 value is the first index of the first
composite number multiple of P1. The t1 value is the first index of the second composite
number multiple of P2. The value of t0 is shifted by -1 with respect to t1 index of the second
composite number multiple of P2.
However, it also requires that p2 indices are written in the form 3m + 2. The indices must be
shifted by 2 compared to the indices of composite numbers multiples of 3.
b) Second linear Diophantine equation
Indices of composite numbers multiple of 3 are written:
Indices of second composite number linked to the pair (P1, P2) are written:
We search the indices p2 such as p2=p0+2 form a pair (p0, p2) that we can write (p, p+2) with
p= p0.
We must solve a new linear Diophantine equation:
Solutions exist because 3 and P1*P2 are coprimes.
The solutions are written in the form a*n + b. It is an arithmetic progression:
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15 formulas that connect twin prime pairs
By replacing y in the formula (2):
With
The value of "a" is equal to the product of the primes. The b1 value is the index of second
composite number multiple of P2.
If b1 is the index of a prime number, we will denote b'1= b1. Then in formula (2) of p2, we
perform a shift by replacing n by n + 1 in order to obtain the index of a composite number
(Property 4-2-1 and Property 4-2-2). The value
is the sought value of b1.
The first pair of composite number is written as (p2-1, p2).
c) The order of the primes in the pair of prime numbers.
The pair (P2, P1) also exists. With 2 primes P1 and P2, there are 2 ways to get a pair of prime
numbers: (P1, P2) and (P2, P1). Thus pairs of composite numbers (p1, p2) created from pair of
primes (P2, P1) are determined as previously by solving 2 linear Diophantine equations.
Solutions are the following indices:
The constants b1 and b2 are different.
Numerically, we are led to the following conjecture:
The two constants are related to prime numbers P1 and P2. Three separate cases exist. The
relationships are different:
Case-1) with a pair of twin primes, when the first pair (6q-1, 6q+1) is written (6q-1=P1,
6q+1=P2). It is not a pair of composite numbers. The indices b1 and b2 of the first pairs of
composite numbers are obtained with the following relationships:
Case-2) with a mixed pair, when the first pair (6q-1, 6q+1) is written (6q-1=P1, 6q+1= y1P2)
or (6q-1=x1P1, 6q+1= P2) with x1, y1 positive natural integers. It is not a pair of composite
numbers. The indices b1 and b2 of the first pairs of composite numbers are obtained with the
following relationships:
Or
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15 formulas that connect twin prime pairs
Case-3) with a pair of composite numbers, the first pair is written (6q-1=x1P1, 6q+1= y1P2)
with x1, y1 positive natural integers. This is the first pair of composite numbers. The indices
b1 and b2 are obtained with the following relationships:
We will demonstrate the formulas obtained for the first 2 cases. We will demonstrate in the
last case that the b1 and b2 results match the criteria of the definition 5.3.2 and then are the
sought solutions.
case-1) With pairs of twin primes, two relations with b1 and b2 exist and depend on P1 and
P2. This determines the values of the b1 and b2 coefficients by solving a system of two
equations with 2 unknowns:
Thus
Property 4-2-1:
The indices (p-1, p) of pairs of composite numbers that are multiple of pair of twin primes
(P1=6m-1, P2=6m+1) or (P2, P1) are written:
The index p of first pair of composite numbers is:
The value of b2 is in the first period
whereas the value of b1 is in the second
period and so a<b1<2*a. Value of indices b is always less than
due to
.
Demonstration:
A pair of twin primes C=(P1=6m-1, P2=6m+1) has the indices c=(p1, p2=p1+1). These indices
only match one of the criteria of the definition 5.3.2 because they are not indices of
composite numbers. The addition of period a = 3*P1*P2 to indices of pair provides the
indices of the first pair of composite numbers that are c1=(p1+3*P1*P2, p2+3*P1*P2) with
b1= p2+3*P1*P2.
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15 formulas that connect twin prime pairs
The indices p1 and p2 are written: p1=(P1-3)/2 and p2=(P2-3)/2.
The pair c1 is then written ((P1-3)/2 + 3*P1*P2, (P2-3)/2 + 3*P1*P2).
We get: b1 = (P2-3)/2 + 3*P1*P2.
The same relation exists in using the first formula of
. We have:
Moreover, b1 is the index of a composite number due to
Value of b1 - 1 is also the index of a composite number:
.
The pair c1 = (b1 - 1, b1) is a pair of indices of a pair of composite numbers. It is written
(3m+1, 3m+2).
The criteria of the definition 5.3.2 are respected.
The first relation for b1 is demonstrated.
Search of the value of b2:
We will retire to previous pair of indices c=(p1=b1-1, p2 =b1) the values of the pair C=(P1,
P2) : d = c - C= (p1-P1, p2-P2) .
The first index becomes greater than the second index:
With P2=P1+2 and p2=b1, p1=b1-1, one gets:
d = (b1-1-P1, b1-P1-2) = (d1, d2)
d1 = d2 + 1
Because d1>d2 then the pair of indices obtained is written by switching the elements of d: d' =
(d2, d1) = (d'1, d'2). The searched value of b2 must be equal to d'2 = d1. Using the formula of
b1 previously obtained, we get:
b2 = d1 = b1-P1-1 = 3*P1*P2 + (P2-3)/2 - P1 - 1
b2 = 3*P1*P2 + (P1+2-3-2P1 - 2)/2
b2 = 3*P1*P2 - (P1+3)/2
Recall: P1 = 6m-1 hence (P1+3)/2 = 3m+1
Index b2 is then written: b2 = 3*P1*P2 - 3m -1=3*(P1*P2 - m) - 1=3m'-1
But with m'=m+1, 3m' - 1 is equivalent to 3m+2.
The difference between the 2 indices of the pair d' is equal to d'2 - d'1=d1 - d2 = 1
The pair d' = (b2 - 1, b2) is then written (3m+1, 3m+2).
The relation of b2 previously obtained can be determined in using the second formula of
We have:
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15 formulas that connect twin prime pairs
Moreover, b2 is an index of a composite number due to
The value of b2 - 1 is also an index of a composite number:
.
The pair d' = (b2 - 1, b2) is a pair of indices of a pair of composite numbers. It is written
(3m+1, 3m+2).
The criteria of the definition 5.3.2 are respected.
The second relation for b2 is demonstrated.
case-2) With a mixed pair that is a prime and a composite number, 2 relations identical to
those of the case 1 may be used. The prime number is either P1 or P2. The mixed pair is then
written as either (P1, P1+2) or (P2-2, P2).
Property 4-2-2:
The indices (p-1, p) of pairs of composite numbers multiples of mixed pair (P1, P2) or (P2,
P1) are:
- when P1 is prime
Index p of the first pair is:
- when P2 is prime
Index p of the first pair is:
The value of b2 is in the first period
and so b2<a whereas the value of b1 is
in the second period and so a<b1<2*a. Value of indices b is always less than
due to
.
Demonstration:
This demonstration is identical to that of the property 4-2-1 by replacing the pair (P1, P2) by
(P1, P1+2) or (P2-2, P2).
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15 formulas that connect twin prime pairs
case-3) When the pairs of primes only generate pairs of composite numbers, indices of
the first pair are written (xP1, yP2) with
. We get 3 relations:
And the following linear equation Diophantine: (1)
Thus with
Or
Hence the following indices representing the set of pairs of composite numbers multiples of
the pairs of primes (P1, P2) and (P2, P1):
The resolution of the linear Diophantine equation leads to determine the particular solution,
that is to say, a pair of integers (x0, y0), such as: (1)
Hence
with
, one gets:
Property 4-2-3:
When the pairs of primes (P1, P2) and (P2, P1) only generate pairs of composite numbers (p1, p), the indices of first pair are written
and
with
natural integers greater than 1.
The resolution of the linear Diophantine equation
leads to determine the
particular solution that is a pair of integers (x0, y0), such as:
.
Let
and
such as:
- when the relation
is true with
, the
solutions are:
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15 formulas that connect twin prime pairs
-
when the relation
solutions are:
s true with
, the
We get then
.
Value of indices b is always less than the period
.
Indices of pairs of composite numbers (p-1, p) obtained with (P1, P2) and (P2, P1) are:
P1 and P2 are two primes whose the pairs (P1, P2) and (P2, P1) does not form a pair (6q-1, 6q
+ 1). They are neither mixed pairs nor pairs of twin primes. The P2 prime is not necessarily
the greatest number of 2 prime numbers P1, P2.
Demonstration:
We will determine the form of integers x1, y1 so that the pair
is written (6q-1,
6q + 1). This requirement matches the criteria of the definition 5.3.2.
We will also determine the mathematical form of x0 so that formulas of b1 and b2 meet the
criteria of the definition 5.3.2.
Composite numbers of the first pair of composite numbers obtained with respectively P1 and
P2 are written (6q-1=x1P1, 6q+1= y1P2) with x1 >1, y1 >1 and
. These integers
exist because P1 and P2 are primes (Property 4-2). The linear Diophantine equation
has a solution. We will determine the mathematical form of values of x1, y1.
In the present case, P1 and P2 are 2 primes which are written as 6q-1 or 6q+1 with
.
There are then 4 separate cases:
case-a) P1 and P2 are written respectively 6
case-b) P1 and P2 are written respectively 6
case-c) P1 and P2 are written respectively 6
case-d) P1 and P2 are written respectively 6
with
.
-1 and 6
+1 and 6
-1 and 6
+1 and 6
+1
-1
-1
+1
The mathematical form of x1 and y1 depend on each previous case. In the first case "a", x1 P1
is written as 6q-1 when x1 is written as 6q1+1 with the natural integer q1>0:
And y1 P2 is written as 6q+1 when y1 is written as 6q3+1 with the natural integer q3>0:
The table below gives the mathematical form of x1 and y1 according to each case.
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15 formulas that connect twin prime pairs
Case
a
b
c
d
P1
6q2-1
6q2+1
6q2-1
6q2+1
P2
6q4+1
6q4-1
6q4-1
6q4+1
x1
6q1+1
6q1-1
6q1+1
6q1-1
y1
6q3+1
6q3-1
6q3-1
6q3+1
(x1 P1)
6q-1
6q-1
6q-1
6q-1
(y1 P2)
6q+1
6q+1
6q+1
6q+1
The pair (x1P1, y1P2) is written (6q-1, 6q+1). The indices of pair are (3m+1, 3m+2).
The criteria of the definition 5.3.2 are respected.
We will determine the mathematical form of x0
h f
f mul f h y m (α1),
case, so that the composite numbers with the indices b1 and b2 are written as 6q + 1
ch
The first formula is written:
In case-a, P1=6q2-1 and P2=6q4+1, we will prove that 2*b1+3 is written as 6q+1.
Let x0 be in the following mathematical form:
with
The relation (1)
leads to
Thus by substitution in the equation (2), one gets:
The calculation of
with the relation (2) leads to:
The pair (2*(b1-1)+3, 2*b1+3) is a pair of composite numbers that are written (6q-1, 6q+1)
multiple of the pair of primes (P1,P2).
The second formula is written:
with
and
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15 formulas that connect twin prime pairs
The table below gives the mathematical form of x0 according to each case. This form is
related to this of P1.
Case
a
b
c
d
P1
6q2-1
6q2+1
6q2-1
6q2+1
P2
6q4+1
6q4-1
6q4-1
6q4+1
x0
3t-1
3t+1
3t-1
3t+1
Examples:
Case
Formula
used
a
a
c
d
Pair (P1,
P2)
(5, 13)
(11, 7)
(5, 11)
(13, 7)
P1
P2
x0
b1
b2
6q2-1
6q2-1
6q2-1
6q2+1
6q4+1
6q4+1
6q4-1
6q4+1
3 t - 1=5
3 t - 1=5
3 t - 1=2
3 t + 1=1
122
59
92
122
71
170
71
149
Note : the values of b1 and b2 are related to the pairs (P1, P2) and (P2, P1). With 2 selected
primes greater than 3, the value taken by P1 and P2 and the formula to use,
,
have not been studied in this article
The second formula is factorized with the factor P1:
Moreover, we get:
The relation (1)
leads to
, hence:
The pair (2*(b2-1)+3, 2*b2+3) is pair of composite numbers that is written (6q-1, 6q+1)
multiple of the pair of primes (P2, P1).
The criteria of the definition 5.3.2 are respected for b1 and b2.
Property 4-3:
Whatever P1 and P2, two primes greater than 3, these numbers are written 6q-1 or 6q+1. A
pair of prime numbers (P1, P2) and its inverse (P2, P1) give rise to pairs of composite
numbers that are written (6q'-1, 6q '+ 1). The indices of these pairs are written (a*n+b1=3m+1, a*n+b=3m+2). The value of "a" is the period equal to
.
One of the two pairs (P1, P2) or (P2, P1) can be:
1- a pair of twin primes
2- a mixed pair
3- neither one nor the other
The value of b represents the index of the second composite number of the first pair of
composite numbers multiple of the pair (P1, P2) or (P2, P1).
Let b1 and b2 be the indices of b related to the 2 pairs of primes.
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15 formulas that connect twin prime pairs
For the first 2 cases, we get the following relation:
The value of b for one of the 2 pairs of composite numbers (b1-1, b1) and (b2-1, b2) is in the
period "a" with b<a and the other is in the second period that is a<b<2*a.
For the last case, we get:
The value of b of the first 2 pairs of composite numbers (b1-1, b1) and (b2-1, b2) is in the
period "a" that is b1<a and b2<a.
Demonstration:
This is deduced from the properties 4-2, 4-2-1, 4-2-2 and 4-2-3.
Property 4-3-1:
When the first pair of odd numbers is a pair of twin primes or a mixed pair, the second pair
obtained by adding the period "a" for each index is a pair of composite numbers. This second
pair is the first pair of composite numbers multiple of the pair of prime numbers. The value of
b is then in the second period that is a<b<2*a.
Demonstration:
The indices of a pair of twin primes (P1=P2-2, P2=6q+1) are:
We let
.
When we add the period a=3*P1*P2 to each index of the previous pair, indices becomes
indices of a pair of composite numbers (X, Y):
Y = 2(b + a) + 3 = P2 – 3 + 6*P1*P2 + 3 = P2*(1 + 6*P1), it is a composite number
X = 2(b – 1 + a) + 3 = P2 – 3 – 2 + 6*P1*P2 + 3 = (P2 - 2) + 6*P1*P2 = P1*(1 + 6*P2), it is a
composite number.
The result is the same with a mixed pair because the composite number of a mixed pair is
written x*P1 or y*P2, with
, depending on its position in the pair. Thus when the
composite number of the mixed pair is in the first position of the pair (xP1=P2-2, P2), indices
are written as
. We get the indices of the first pair of composite numbers (X, Y) in
adding the period "a" to each index:
Y = 2(b + a) + 3= P2*(1 + 6*P1), it is a composite number
X = 2(b – 1 + a) + 3= (P2 – 2) + 6*P1*P2=P1*(x + 6*P2), it is a composite number.
A similar result is obtained when the first pair (6q'-1, 6q'+1) multiple of the pair of primes
(P1, P2) is a mixed pair (P1, yP2=P1+2).
Property 4-3-2:
The pairs of primes (P2, P1) reciprocals of twin prime pair (P1, P2) and mixed pair (P1, P2)
generate pairs (6q'-1, 6q'+1) that are only pairs of composite numbers. The value of index b of
the first pair of composite numbers (b-1, b) is always less than the period a = 3*P1*P2: b<a.
Demonstration:
A pair of twin primes is written (P1=6q-1, P2=6q+1). The reverse pair of prime numbers (P2,
P1) generates the first pair of composite numbers (6q’-1, 6q’+1) in multiplying primes P1 and
P2 with factors such as:
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15 formulas that connect twin prime pairs
xP2 = x (6q+1) = 6q’-1
yP1 = y (6q-1) = 6q’+1
with
. The determination of x and y values is made by solving two
linear Diophantine equations (Property 4-2).
A mixed pair is written (xP1=6q-1, P2=6q+1) or (P1=6q-1, yP2=6q+1), with
. The reverse pair of primes (P2, P1) generates the first pair of composite
numbers (6q’-1, 6q’+1) in multiplying primes P1 and P2 with factors. In the first case
(xP1=6q-1, P2=6q+1), the reverse pair (P2, P1) creates the pair (6q’-1, 6q’+1) :
z P2 = z (6q+1) = 6q’-1
t*x*P1 = t (6q-1) = 6q’+1
with
. The determination of values of t*x and z is made by solving
two linear Diophantine equations (Property 4-2).
The result is similar with the second case.
Remark:
The indices (p-1, p) of pairs of composite numbers multiples of pairs of primes are written as
arithmetic progression p=a*n + b. Values of "a" and "b" are obtained by using the extended
Euclidean algorithm.
Example:
Indices (p2-1, p2) of pairs of composite numbers multiples respectively of the pairs of primes
(5,7) and (7,5) are represented by the following formulas:
Pair of primes
Pair of indices of pair of composite numbers depending on n
n =0
n =1
n =2
n =3
n =4
(5,7)
(106,107)
(211,212)
(316,317)
(421,422)
(526,527)
(7,5)
(100,101)
(205,206)
(310,311)
(415,416)
(520,521)
Here are the formulas p2 for several pairs of primes (P1, P2) and (P2, P1).
Pair of primes
[5,7]
[5,11]
[7,11]
[5,13]
[7,13]
[11,13]
[5,23]
p2
107+105*n
92+165*n
59+231*n
122+195*n
122+273*n
434+429*n
332+345*n
Reverse pair
[7,5]
[11,5]
[11,7]
[13,5]
[13,7]
[13,11]
[23,5]
p2
101+105*n
71+165*n
170+231*n
71+195*n
149+273*n
422+429*n
356+345*n
V.3.3 Enumeration and computing value of C1
Indices (3m+1, 3m+2) of composite numbers (6q-1, 6q+1) are written depending on primes
(P1, P2). These indices are written as arithmetic progressions:
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15 formulas that connect twin prime pairs

with a = 3*P1*P2. We will denote
with i being the number of primes
greater than 3 in combination that is to say in this example the pair (P1, P2), and
with Nmax being the number of
primes greater than 3 in the interval [0,

].
with b the index of the second composite number of first pair of composite numbers
multiples of pair of primes (P1, P2). Value of b depends on primes P1 and P2. We will
write it:
.
The number of pair of composite numbers related to pair (P1, P2) is obtained with indices p2.
In an interval [0, Np], with Np an odd number with index p, the number of pairs is computed
with the following relationship:
When p is equal to
Hence the relation:
, number of pairs is equal to 1.
For a selected odd number Np, the existing composite numbers in the interval [0, Np] are
multiples of the existing primes in the interval [0,
]. Enumeration of pairs of composite
numbers is performed using prime numbers present in the preceding interval. The number of
prime numbers is equal to Nmax + 1. The first odd prime is P0=3 and the last one is PNmax.
The number of pairs of primes is equal to the following permutation:
In the interval [0, Np], the number of pairs of composite numbers multiples of a pair of primes
is obtained with all pairs of primes by the relation below:
with Pjkm that is primes of a combination k of the permutation
and
.
is a constant that is index of the second composite number of the first pair of composite
numbers multiples of pair of primes (Pjk1, Pjk2).
is the product of primes present in the combination. Value of "i" is the number of
primes in the combination that is equal to 2 with one pair of primes.
The enumeration realized for C1 is to count each pair of composite numbers only once.
Indices (3m + 1. 3m + 2) of some pairs of composite numbers (6q-1, 6q + 1) can be written as
depending on different pairs of prime numbers (P1, P2), (P3, P4) and so on. These pairs of
composite numbers are then counted several times in c1. We call c2 the number of times that
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15 formulas that connect twin prime pairs
a pair of composite numbers is counted more than 1 time. The sought value of the parameter
C1 is equal to
.
Enumeration for c2:
A pair of composite numbers multiples of pairs (P1, P2) and (P3, P4) will be counted twice in
the formula c1. The first composite number is then multiple of primes P1 and P3 while the
second one is a multiple of P2 and P4. We must retire the number of times that the 2 pairs are
present for the same pair of composite numbers. We denote that the pairs of primes are
common to a same pair of composite.
However, the number of pairs of prime numbers common to pairs of composite numbers may
be greater than 2. This number is related to the number of existing pairs equal to
.
Definitions 5.3.3.1:
1) We will call NCmax the maximum number of pairs of prime numbers common to a same
pair of composite numbers.
2) We will denote NC the number of pairs of prime numbers common to a same pair of
composite numbers
.
The number of times that two pairs of prime numbers are common to a same pair of
composite numbers must be retired to enumeration. It is the same every time the number NC
of pairs of primes is even.
The number of times that 3 pairs of prime numbers are common to a same pair of composite
numbers must be added to enumeration. It is the same every time the number NC of pairs of
primes is odd.
Definitions 5.3.3.2:
1) The addition (+) or removing (-) of values to the enumeration is determined by the
relationship: signNC = (-1)NC+1.
2) The number of pairs of primes depends on the number of primes equal to Nmax. This
number is equal to the value of the mathematical combination below:
The combinations of the pairs of primes are not all valid because one cannot find the index of
a composite number that is multiple of prime number in both 3m+1 and 3m+2 (See Property
4-1).
3) Indices (p1, p2) of pairs of composite numbers multiples of 2 pairs of primes (P1, P2) and
(P3, P4) are a subset called
of the sets
and
such as:
This subset represents the indices (p1, p2) in common with those of sets
and
.
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15 formulas that connect twin prime pairs
We denote NF2(Ci)=card(
) with "i" the number of distinct primes,
C4={P1,P2,P3,P4} with
. When the prime numbers of 2 pairs are distinct from each
other, the value of "i" is equal to 4. If a value is common, for example P1 = P3, the value of
"i" is 3 and C3 = {P1, P2, P4}. Thus, we can write:
with NC number of pairs of
primes. With 2 pairs, we get:
.
For a number of pairs of primes equal to NC composed of a number of distinct primes equal
to "i", NF2(Ci) is the number of times that these pairs of primes are common to a same pair of
composite numbers in interval [0, Np]. These pairs of primes form a combination.
with NC the number of pairs of primes of the combination which contains "i" primes with
.
with
and so
and
.
Let
be the set of primes present in first position of the pairs:
and
And let
be the set of primes present in second position of the pairs:
and
The valid combinations meet
Numerical example of combinations of pairs of primes obtained with up to 4 pairs of prime
numbers: we remark that a combination of four pairs of primes cannot exist because the index
of a number that is multiple of 7 cannot be in both 3m + 1 and 3m + 2. Combinations
including the 2 pairs (5. 7) and (7. 11) are not taken into account because the pairs of
composite numbers formed from these combinations do not exist.
Number of pairs of
primes (NC)
Add/Remove
values
1
+
2
-
NC=1, pairs of primes
NC>1, valid combinations of pairs of primes
(5,7)
(5,11)
(7,11)
(5,13)
(5,7) (5,11)
(5,11) (7,11)
(5,7) (5,13)
(5,11) (5,13)
(7,11) (5,13)
3
+
(5,7) (5,11) (5,13)
(5,11) (7,11) (5,13)
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15 formulas that connect twin prime pairs
Determining the value of
Definitions 5.3.3.3:
:
1. Let a combination be constituted of a number of pairs of primes equal to NC with
NC> 1. Every pair of primes creates a sequence of pairs of composite number. We
seek pairs of composite numbers common to pairs of composite numbers generated by
each pair of prime numbers. Such a combination consists of two sequences of indices
(p1, p2=p1+1).
2. A valid combination of two or more pairs of primes is determined by an equality of p2
index of the second composite numbers of pairs of composite numbers multiples of
pairs of primes.
Property 4-3-3: A valid combination of two or more pairs of primes always has a solution
corresponding to an arithmetic sequence. The values of the index p2 of the second composite
number of the sought pairs of composite numbers are written:
such as:
 with "a" equal to 3 times the product of the distinct primes from each other of the
combination. One denote
=
with
and Pj being
distinct primes of the combination. For example, with two pairs (P1, P2) and (P3, P2), the
value of "a" is equal to 3*P1*P2*P3. A combination which gathers a number of pairs
equal to NC consists of a number of distinct prime numbers equal to at most i=2*NC.
 with "b" that is the index of the second composite number of the first pair of composite
numbers and it will be written
.
The values of indices of b1 and b2 of a valid combination and its inverse combination are
connected by the following relationship:
and Pj being
the distinct primes of the combination. One then gets:
Demonstration:
Valid combinations are obtained with a number of pairs of primes NC with
. Let
NC=2 and a combination with the 2 pairs (P1, P2) and (P3, P4).
Indices of pairs of composite numbers multiples of respectively prime numbers P1 and P2 of
the first pair of prime numbers (P1, P2) are written:
.
Indices of pairs of composite numbers multiples of respectively prime numbers P3 and P4 of
the second pair of prime numbers (P3, P4) are written:
.
The indices in common are obtained by solving a linear Diophantine equation:
The solutions correspond to an arithmetic sequence:
The combination (P1, P2), (P3, P4) and its inverse combination (P2, P1), (P4, P3) have the
same value "a" but different values of b that are b1 and b2 with
.
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15 formulas that connect twin prime pairs
Valid combinations consisting of a number of pairs of primes equal to NC separate into 2
categories:
1- In the first category, the prime numbers of the pairs of a combination are all different.
2- In the second category, two or more prime numbers are identical in pairs of primes of the
combination.
1- The combinations of the first category lead numerically to the following relationships
between the values b1 and b2.
If we set
and
, then we get the same relationships as that of
the property 4-2-3. This property gives the relationship of b1 and b2 according to the prime
numbers in the case where the pairs of prime numbers (P1, P2) and (P2, P1) only generate
pairs of composite numbers (6q-1=
, 6q+1=
) and (6q’-1=
, 6q’+1=
). We
will prove that formulas of b1 and b2 obtained from three previous relationships meet the
criteria of the definition 5.3.2. They represent the second index of the first pair of composite
numbers obtained for the combination studied and the reverse combination.
Property 4-3-3-1:
Valid combinations that consist of NC pairs of primes only generate pairs of composite
numbers. The primes that compose pairs of primes are all different from each other. The pairs
of primes are denoted (P1H, P2H) with
. We denote
and
. The indices of the first pair of composite numbers are written
and
with
natural integers greater than 1.
The resolution of the linear Diophantine equation
leads to determine the
particular solution that is a pair of integers (x0, y0) such as
.
We denote
and
such as:
- When the relation
is true with
, the
solutions are:
-
When the relation
solutions are:
Then we get:
is true with
, the
.
Demonstration: This demonstration is identical to that of the property 4-2-3 in using P1=P1’
and P2=P2’.
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15 formulas that connect twin prime pairs
2- The combinations of the second category lead numerically to the following
relationships between the values b1 and b2 when P1=P3 :
We will prove that formulas of b1 and b2 obtained from three previous relationships meet the
criteria of the definition 5.3.2. They represent the second index of the first pair of composite
numbers obtained for the combination studied and the reverse combination.
Property 4-3-3-2:
Valid combinations that consist of NC pairs of primes only generate pairs of composite
numbers. Some of primes that compose pairs of primes are identical. The pairs of primes are
denoted (P1H, P2H) avec
. The set of prime numbers in the first and second
position of pairs are respectively noted:
The set of unique primes in first and second position of pairs are respectively noted:
We denote
and
. The indices of the first pair of
composite numbers are written
and
with
natural integers greater than 1.
The resolution of the linear Diophantine equation
leads to determine the
particular solution that is a pair of integers (x0, y0) such as:
.
We denote
and
such as:
- When the relation
is true with
,
the solutions are:
-
When the relation
the solutions are:
is true with
,
Then we get:
.
When we write P1'=P2'' and P2'=P1'' that is to say we reverse the elements of the pairs, we get
the following relationship:
. The formulas of b1 and b2 are then those
obtained with the property 4-3-3-1.
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15 formulas that connect twin prime pairs
Demonstration: this demonstration is similar to the property 4-2-3 in using P1=P1’’ and
P2=P2’’. Details of this demonstration are available in Appendix 1.
The number of indices in common is then written
. However, when p = b(Ci),
number of indices is equal to 1. Hence:
The value of c2 is then equal to:
with k representing a valid combination consisting of NC pairs of primes taken from a number
of pairs equal to
.
with b(Ci) that is a constant that depends on primes P present in the pairs of the combination.
The formula for c1 is similar to the formula for c2 with:
- NC=1 and so
=(-1)2 = 1
- i is equal to 2
Thus C1 = c1 + c2 is written as in property 4-4 below:
Property 4-4:
The number of pairs of composite numbers C1 existing in the interval [0, Np] is determined
from the number of pairs of prime numbers present in the interval
that is equal to
NCmax. These pairs are created from the existing primes in the interval
whose
their number is equal to Nmax+1.
with k representing a valid combination consisting of NC pairs of primes taken from a number
of pairs equal to
.
with b(Ci) that is a constant that depends on primes P present in the pairs of the combination.
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15 formulas that connect twin prime pairs
V.3.4 C1(j) is a strictly increasing function
Index of an odd number Np is equal to p. The enumeration for C1 is conducted between 0 and
the square of an odd number. The index of the square of an odd number is called remarkable
number. This number is written depending on j such that:
The enumeration is performed for each sequence
up to remarkable number of the next
sequence
. In other word, for a selected sequence , enumeration is performed in the
interval
.
Function C1(p) is written according to the parameter j if we replace p by
in
formula of the property 4-4. C1(j) is a cumulative function. When value of j increases, Np
also increases, hence C1(j+1) ≥C1(j). When adding to C1 the number of pairs of composite
numbers that are multiple of a new pair of primes, only those non common to previous ones
are counted. The C1 value can therefore only grow with the prime numbers
.
We will demonstrate that the number of pairs of composite numbers is strictly increasing with
j from j = 3.That is to say:
.
The first pair of prime numbers greater than 3 is the pair (5, 7).
The number of pairs of composite numbers multiple of the pair of primes (5, 7) is equal to
C15,7 which is the is the cardinality of the following set (See paragraph V-3-1):
with C15,7 = card(
)
C15,7 = c1 + c2 with c2 = 0 because there is only 1 pair and so NC=1.
C15,7 = c1 =
Let p = 3 + 105*n. This choice will be explained in paragraph VII-1.
When p = 3 + 105*n, C15,7 = n
There is at least one pair of composite numbers every 105 consecutive indices. The number of
indices in a base unit is equal to d=4j+7. The number of indices in a base unit exceeds 105
from j = 25. The numerical values of C1 present in the table below show a growth from j = 3.
The value of C1 increases as a function of j from the value j = 3.
J
C1
d
0
0
7
1
0
11
2
0
15
3
1
19
J
C1
d
13
34
59
14
40
63
15
46
67
16
56
71
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4
2
23
17
64
75
5
5
27
18
71
79
6
7
31
19
82
83
7
10
35
20
93
87
8
12
39
9
15
43
21
107
91
10
19
47
22
116
95
WOLF Marc, WOLF François
11
23
51
23
130
99
12
28
55
24
25
140
153
103 107>105
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15 formulas that connect twin prime pairs
Property 4-5:
The number of pairs of composite numbers C1(j) is a strictly increasing function from j = 3:
V.3.5 Evolution of C1(j) to infinity
We will demonstrate that C1(j) evolves as a second degree polynomial function.
In the interval
, the number of primes greater than 3 is equal to Nmax. The number
of existing pairs of composite numbers C1 in the interval [0, Np] is equal to:
A remarkable number is written:
The enumeration is performed for each sequence
up to remarkable number of the next
sequence
. In other word, for a selected sequence , enumeration is performed in the
interval
.
By substituting
in the formula C1(p), we get:
When j is equal to a multiple of the lowest common multiple (LCM) between all PF2( ) and
3, C1(j) is written as a second degree polynomial function. The LCM(PF2( ),3) is equal to
the product of all prime numbers
.
, C1 is written according to
.
Property 4-6:
When the number of primes (Nmax) tends to infinity, the number of pairs of primes (NCmax)
also tends to infinity. The following sum
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15 formulas that connect twin prime pairs
Is equal to at least 4 less the number of pairs of twin primes (
pairs that is called NCM. We then get:
) and the number of mixed
Demonstration:
Recall: two primes give rise to two pairs of prime numbers (P1, P2) and (P2, P1). Each of
these pairs gives rise to pairs of composite numbers (6m-1, 6m + 1). The indices of these pairs
(p-1, p) are written as an arithmetic sequence:
. For n=0, the indices p
of each pair are b1 and b2. They are the indices of the first pair of composite numbers (6m-1,
6m+1).
Let
be as follows:
The value of Va is equal to 0 if
The value of Va is equal to -1 if
It has been shown that
2-2 and property 4-2-3). We get
.
.
for every pairs (property 4-2-1, property 4.
It has been demonstrated (property 4-2-3) that for a pair of primes of which the first pair
written as (6q-1, 6q+1) is a pair of composite numbers, the value of b(Ci) is less than
.
The value of Va for a pair of composite numbers is then always equal to 0.
It has been demonstrated (property 4-2-1 and property 4-2-2) that for a mixed pair or a pair
of twin primes, one of values of b(Ci), b1 or b2, is greater than
and the other is
inferior. The values b1 and b2 are related by the relationship below:
For the pairs of twin primes, the value of
greater than
is obtained by
formula:
The value of Va is equal to 0 for the 3 first pairs (5, 7), (11, 13) and (17, 19)
The value of Va is equal to -1 for the other pairs because
Then the partial sum
of obtained for the pairs of twin primes is equal to 1= 3 -
.
For the mixed pairs, when the first number P1 of the pair (P1, P2) is the prime number, the
value of
is equal to:
Otherwise when the second number P2 of the pair (P1, P2) is the prime number, the value of
is equal to:
The value of Va is equal to 0 with only the mixed pair (P1=23, 5*P2=25) that is the first pair
(6q-1, 6q+1) obtained from the pair of primes (23, 5). A value of Va equal to 0 is obtained
only for the 2 previous cases when respectively the prime number:
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15 formulas that connect twin prime pairs
- P1 of the mixed pair is less than or equal to 23
- P2 of the mixed pair is less than or equal to 23 because 25 is not prime.
The value of Va is equal to -1 for other mixed pairs.
Then the partial sum
of obtained for the mixed pairs is equal to = 1 - NCM with NCM
the number of mixed pairs.
Note:
The numerical results obtained with the first 25 prime numbers greater than 3, show that only
the mixed couple (23, 25) leads to a value of 0. The other mixed pairs are created with a prime
number greater than 23. The primes less than 23 form pairs of twin primes (5, 7), (11, 13) and
(17, 19).
The sum
is then equal to:
Property 4-7: Let the following relationship:
When j tends to infinity, the number of primes (Nmax) and so the number of pairs of primes
also tends to infinity. The series aC is a convergent special alternating series. It converges to
1/3:
The following special alternating series converges to 1:
Demonstration: using the precedent properties allows us to write the relationship:
Using the formula
, we get:
But
Hence the limit to infinity of
:
The value of aC is written as an alternating series linked to the alternation of the sign
(-1)(NC+1). But this is a special series because of the term of the series which depends on NC
and Nmax. It is a double sum. The term is written as follows:
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15 formulas that connect twin prime pairs
The demonstration is done.
Remark: In the table below, values of aC are given according to the number of primes Nmax
up to the first 8 prime numbers. For Nmax=8, the coefficient aC is calculated with all terms of
the series up to the 16th term, called U16. The calculation is also done for the first 6 and 7
terms that are
and
.
Nmax
2
3
4
5
6
7
8
aC (16, Nmax)
0,01904762 0,03463203 0,04861805 0,05954829
0,0693651 0,07744721 0,08381258
aC (6, Nmax)
0,01904762 0,03463203 0,04861805 0,05954829 0,06920634 0,07656652
aC (7, Nmax)
0,01904762 0,03463203 0,04861805 0,05954829 0,06940428 0,07780582 0,08518424
0,0812351
Illustration 7:
0,09
0,08
0,07
0,06
0,05
ac (16)
0,04
ac (6)
0,03
ac (7)
0,02
0,01
0
2
3
4
Nmax 5
6
7
8
C1(j) is a strictly increasing function. Its value is equal to that of a second degree polynomial
(A1) of which pairs of twin primes and mixed pairs are removed:
. Since there are
infinitely many prime numbers, the value of
, named N1, is also infinite. With
14
Np=10 -1,
= 135780321664. The N1 value is strictly between /2 and and so when j
tends toward infinity:
The curve obtained from a calculation up to a value greater than Np=1014+1, is presented
below. This curve shows that C1 can be represented by a second degree polynomial function.
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15 formulas that connect twin prime pairs
Note: the first two coefficients of second degree polynomial are depending on the parameter
"j" because they are related to the infinity of prime numbers that are written:
. When j tends to infinity, the first coefficient increases toward 2/3 and the
second coefficient decreases to minus infinity because the sum
is infinite.
Illustration 8: the graph of equation C1 = f(j) (Trend curve) and the regression coefficient R
were obtained with the Maple tool. The data are in black color and the trend curve is in red
color. The calculation was made until j = 5 000 170. For j = 4 999 998, the odd number Np =
1014 +1 is reached.
C1(j) = 0.548335414807537 j2 - 23914.4178108798 j
R = 0.999999584475459
VI- The infinity of pairs of twin primes
VI.1 Enumeration of pairs of twin primes
We have shown that the number of pairs of twin primes
= A1(j) - B1(j) + C1(j).
The value of
in an interval [0, Np] is written:
is written depending on j:
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15 formulas that connect twin prime pairs
The lowest common multiple (LCM) between all
product of all prime numbers
.
When j = LCM (3,
,
, all PF( ) and 3 is equal to the
) * n, we get:
= –(
+NCM)
= ( - NCM) / 2
Illustration 9:
The number of pairs of twin primes obtained in the interval [0, 10t] with
is an
integer sequence referenced in the On-Line Encyclopedia of Integer Sequences (OEIS) as
A007508. Our calculation is performed only with pairs of numbers (6m-1, 6m + 1). The pair
(3, 5) is excluded. So we get the same values less 1.
The number of pairs of twin primes was calculated depending on j. The curve below shows
that
can be represented by a second degree polynomial function.
Note: the first two coefficients of second degree polynomial are depending on the parameter
"j" because they are related to the infinity of prime numbers that are written:
.
The graph of equation
(Trend curve) and the regression coefficient R were
obtained with the Maple tool. The data are in black color and the trend curve is in red color.
The calculation was made until j = 5 000 170. For j = 4 999 998, the odd number Np = 1014 +1
is reached.
(j) = 0.00500308229559492 j2 + 2293.67889410006 j + 3
R = 0.999967565524202
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15 formulas that connect twin prime pairs
VI.2 New conjecture on twin primes
We demonstrated that the functions A1, B1 and C1 are all strictly increasing according to j.
Legendre's conjecture tells us that the number of primes
is also a strictly
increasing function according to j. That is to say that the number of primes between the square
of two consecutive odd numbers is equal to at least 2.
We calculated the number of pairs of twin primes according to j. The calculation was made
until j = 5 000 170. For j = 4 999 998, the odd number Np = 1014 +1 is reached. The
calculation shows that the number of pairs of twin primes is a strictly increasing function until
at least j = 5 000 170.
Note: The computation of the number of primes was also performed. It shows that there are at
least two prime numbers for each value of j and this at least until j = 5 000 170.
Conjecture :
We state that there is at least one pair of twin primes between the square of two consecutive
odd numbers that is to say between n2 and (n+2)2 with n an odd number.
VII- Formulas connecting the twin prime pairs
It is known that the set of prime numbers is represented by the following two mathematical
formulas with the exception of the integers 2 and 3:
Np = 6m-1 or Np = 6m+1 with m>0
This corresponds to exclude the integers multiples of 2 and multiples of 3.
The reciprocal is not true.
Currently, there is no equivalent on the set of pairs of twin primes that are written as (6m-1,
6m + 1). Only the pair of twin prime numbers (3, 5) cannot be written in this form.
We will show that there are 15 formulas that connect all pairs of twin primes except for pairs
(3, 5) and (5, 7). The reciprocal is not true.
VII.1 Combination of 3 sequences W0, W1 and W2
The pairs of integers (6m-1, 6m + 1) included all pairs of twin primes except the first pair (3,
5). The integers that form these pairs are located between the integers multiple of 3. Between
two integers multiples of 3, there are only two odd numbers. Odd numbers that follow the
integer 3 are 5 and 7. These two integers form the first pair of twin primes (5, 7) written as
(6m-1, 6m + 1).
We will calculate all pairs written as (6m-1, 6m + 1) where the odd numbers 6m-1 and 6m + 1
are neither multiple of 5 nor multiple of 7. For this, we will use the first 3 sequences W0, W1
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15 formulas that connect twin prime pairs
and W2. All combinations of the three sequences are in an interval of length equal to 3*5*7 =
105 (Property 13). The natural period is equal to 105. The first index p of the sequence W0 is
3. The combinations of the 3 sequences are repeated every 105 consecutive indices. One then
finds the first index of each combination in the following interval:
and then the
next indices every 105 consecutive indices that is
,
.
We will determine how many pairs of twin primes may exist at maximum when p = 3 +
105*n that is to say the number of pairs of odd numbers whose values are neither multiple of
5 nor multiple of 7. These pairs will be named « pairs of virtual twin primes ».
VII.2 Calculation of the number of pairs of virtual twin primes
Property 5-1:
The pair (5, 7) exists when p = 2. From p = 3, every 105 consecutive indices, there are only 15
pairs of odd numbers that may be pairs of twin primes. These pairs are called « pairs of virtual
twin primes ».
Demonstration:
The number of pairs of twin primes
is written:
= A1(p) – B1(p) + C1(p)
We must calculate the number of pairs of odd numbers in which none of the two numbers are
multiples of one of these primes: 3, 5 and 7. These pairs are defined as pairs of virtual twin
primes. The calculation is performed with the following first 3 primes: P0 = 3, P1=5 and P2 =
7. The value of Nmax is equal to 2 and the number of pairs NCmax is 1. The values of B1 and
C1 calculated only with the 3 precedent primes are partial values B1' and C1'. Indeed, the
calculation is not performed with the set of prime numbers present in a interval [0, Np] with
Np an odd number.
The values of A1, B1 and C1 depending on index p are given by the relations below:
Property 2-1:
The number of existing pairs of indices (3m+1; 3m+2) in the interval [0, Np] with Np an odd
number written as Np=2*p+3 is obtained by the following relation:
Property 3-2:
The number of existing composite numbers B1 in the interval [0, Np] is determined from the
prime numbers present in the interval
. Number of primes is equal to Nmax+1.
With
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15 formulas that connect twin prime pairs
Property 4-4:
The number of pairs of composite numbers C1 existing in the interval [0, Np] is determined
from the number of pairs of prime numbers present in the interval
that is equal to
NCmax. These pairs are created from the existing primes in the interval
whose
their number is equal to Nmax+1.
with k representing a valid combination consisting of NC pairs of primes taken from a number
of pairs equal to
.
with b(Ci) that is a constant that depends on primes P present in the pairs of the combination.
When p=3+105*n, the values of A1, B1’ et C1’depending on “n” are:
A1(n)
35n+1
B1’(n)
22n
= 21n + 15n - 3n - 5n - 7n + n
C1’( )
2n
=n+n
A1(n) - B1’(n) + C1’(n) =
1 + 15n
Thus, the number of sought pairs of odd numbers (6m-1, 6m + 1) equal to the number of pairs
of indices (3m + 1, 3m + 2) is equal to
A1(n) - B1’( ) + C1’( ) = 1 + 15n
With the value 1 that is the pair (5, 7).
VII.3 Theorem: 15 formulas connecting the pairs of twin primes
From p = 3, every 105 consecutive indices, there are 35 pairs (3m + 1, 3m + 2) that are 35
pairs of composite numbers (6m-1, 6m + 1). Among them, 20 pairs (6m-1, 6m + 1) have at
least one number that is multiple of 5 and / or multiple of 7. There are 15 pairs (6m-1, 6m + 1)
whose values 6m-1 and 6m + 1 are neither multiple of 5 nor multiple of 7.
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15 formulas that connect twin prime pairs
Whatever the selected index in the range [3, 107], if the associated odd number factorizes
with 5 and / or 7, it will also be every 105 consecutive indices. Similarly, if the odd number
associated with the index named px, is factorized neither by 5 nor by 7, this will be true for
every indices which are written as:
Theorem:
There are 15 formulas that connect twin prime pairs except for the first 2 pairs (3, 5) and (5,
7). The reciprocal is not true.
The 15 formulas that connect twin prime pairs are given in the table below.
The formulas below give the index and the value of the first virtual prime number of a pair of
virtual twin primes.
Let p, the index of the first virtual prime
number of a pair of virtual twin primes
Let Np, the first virtual prime number of a
pair of virtual twin primes
p=4+105*n
Np=11+210*n
p=7+105*n
Np =17+210*n
p=13+105*n
Np =29+210*n
p=19+105*n
Np =41+210*n
p=28+105*n
Np =59+210*n
p=34+105*n
Np =71+210*n
p=49+105*n
Np =101+210*n
p=52+105*n
Np =107+210*n
p=67+105*n
Np =137+210*n
p=73+105*n
Np =149+210*n
p=82+105*n
Np =167+210*n
p=88+105*n
Np =179+210*n
p=94+105*n
Np =191+210*n
p=97+105*n
Np =197+210*n
p=103+105*n
Np =209+210*n
The reciprocal is not true. All indices or all odd numbers obtained with the formulas above do
not lead to a pair of twin prime numbers.
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15 formulas that connect twin prime pairs
VIII- Infinity of pairs of virtual twin prime numbers
We will prove that the number of pairs of virtual twin primes is infinite when the number of
prime numbers tends to infinity. Moreover, this number grows strictly with the number of
prime numbers.
The preceding theorem says us that there are 15 formulas which connect the first prime
numbers of pairs of twin prime numbers with the exception of the first two pairs (3, 5) and (5,
7). The reciprocal is not true. These formulas make it possible to obtain the pairs of twin
prime numbers. There are fifteen formulas that connect the first prime numbers and fifteen
formulas that connect the second prime numbers of the pairs of twin prime numbers. Some of
the indices obtained from the thirty formulas exist in the set W since they are the indices of
composite numbers. We will determine the number of these indices when the natural period
increases with the number of prime numbers. The evolution of the number of pairs of virtual
twin primes is then determined.
VIII-1- Definitions
Recall: in the set W, natural period is obtained with the first "n" prime odd numbers. Its value
is the product of these prime numbers. We will note the natural period .
Definition 8-1-1:
1. In the set of natural integers, natural period is defined with the first "n" prime
numbers. The even prime number “2” is included. The value of the natural period is
got by the product of these prime numbers. We will note this period
.
2. In the set W, we denote formula (A) that is one of the fifteen formulas which connect
the index of the first prime number of pairs of twin prime numbers. We will denote
formula (B) that is one of the fifteen formulas that connect the index of the second
prime number of pairs of twin prime numbers. Each formula (A) is associated with a
formula (B) to form the indices of pairs of odd numbers such that:
(B) = (A) + 1.
3. In the set W, we denote formula (A') that is one of the formulas which connect the first
prime number of pairs of twin prime numbers. We will denote formula (B') that is one
of the formulas that connect the second prime number of pairs of twin prime numbers.
Each formula (A') is associated with a formula (B') to form the pairs of odd numbers
such that: (B') = (A') + 2.
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15 formulas that connect twin prime pairs
VIII-2- Determination of the number of pairs of virtual twin numbers in a
natural period
Property 8-2-1:
Let the first "z + 1" prime odd numbers identified by an index ranging from 0 to z, we denote
by Fz the zth prime number in ascending order and greater than 3. We set U0 the first term of a
particular progression and U0 = 15. The value of this term is equal to the number of pairs of
virtual twin prime numbers existing in the natural period equal to
with z=3.
The number of pairs of virtual twin prime numbers present in a natural period equal to
is represented by a particular progression
with
and:
This progression
is written:
After simplifying:
The number of pairs of virtual twin prime numbers, called Un, tends to infinity when the
number of prime numbers
tends to infinity. So with z = n + 3, we get:
Demonstration:
The fifteen Formulas (A') were determined from the following prime numbers 3, 5 and 7. The
first fifteen odd numbers exist in the natural period
. They are in the
interval [9; 217]. These fifteen odd numbers are multiples of none of the following three
factors: 3, 5 and 7.
In the set W, we work with the indices of odd numbers. The first fifteen indices were
determined in the period
. They are in the interval [3; 107]. When the
interval increases by each 105, fifteen new indices are determined from the fifteen formulas
(A).
Each odd number and its index given by one of the formulas (A') and (A) respectively
represents the first virtual prime number and its index of a pair of virtual twin prime numbers.
When one of the two odd numbers that form the pair of virtual twin prime numbers is multiple
of a prime number greater than 7, that pair must be removed from the pairs of virtual twin
prime numbers. In an interval equal to the natural period
, there are two pairs to be
2
removed: (167, 169=13 ) and (209=11*19, 211). These values are obtained with n = 0 for the
11th and 15th formula of pairs of odd numbers. There are only 13 pairs of twin prime
numbers in the interval [9, 217].
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15 formulas that connect twin prime pairs
We will seek for the number of pairs of virtual twin prime numbers that exist when the natural
period increases. The natural period of the indices is equal to the product of the first "n" prime
factors. With the first three prime numbers, the natural period is equal to 105 and there are
fifteen formulas. With the first four prime numbers, the factor 11 is added to the list of prime
numbers studied and the natural period is equal to 1155. The interval is then [3, 1157]. This
interval contains eleven times the previous interval equal to 105. The number of pairs of
virtual twin numbers is then equal to 15 * 11 = 165. However, it is necessary to remove the
number of pairs whose at least one of the odd numbers is multiple of the factor 11.
Note: two consecutive odd numbers cannot be multiple of the same odd factor. The difference
between two consecutive odd numbers is equal to two and the minimum difference between
two consecutive odd numbers multiples of the two smallest odd factors, which are 3 and 5, is
equal to 6, i.e. 3 * 5 - 3 * 3 = 6. The indices of two odd numbers that are multiples of 11
cannot be found on both formula (A) and its associated formula (B).
We will seek for the odd numbers that are multiple of 11 and belonging to fifteen formulas
(A) and fifteen formulas (B). We will work with the indices of odd numbers. The indices of
the odd numbers multiples of 11 are given by the sequence Wj with j=4 and Fj=2j+3=11:
a) The first one of the fifteen formulas (A) is:
We shall find the values of k(n) which satisfy the following criterion:
in order
to obtain the indices of the odd number multiples of 11 and belonging to the first formula.
So we must solve a Diophantine linear equation:
A solution exists because the values 11 and 105 are coprime integers. The solution is obtained
by using
. The results are related to an arithmetic progression
with common difference equal to the natural period that is 1155:
b) The first one of the fifteen formulas (B) is:
So we must solve a Diophantine linear equation:
The solution is obtained by using
. The results are
related to an arithmetic progression with common difference equal to the natural period that is
1155:
In an interval equal to 1155, there are two indices of odd numbers that are multiples of 11 and
belonging to the first formula (A) and (B). So there are two pairs to be removed because at
least one of the two odd numbers of these two pairs is multiple of 11. However, in the interval
[3, 1157], the index of value 4 is the index of the first prime number of a pair of twin prime
numbers. This pair, named Cp, is to be preserved. Then there is only (2-1) = 1 pair to be
removed. This result exists whenever the prime factor, which forms the new natural period,
belongs to a pair of twin prime numbers. The index of the prime factor is then equal to one of
the indices of formulas (A) and (B).
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15 formulas that connect twin prime pairs
The other fourteen virtual twin prime pairs lead to solve 28 Diophantine linear equations. For
each of the other 28 formulas (A) and (B), only the constant of the equation is different. So
there is always a solution. In an interval equal to 1155, we get 28 indices of odd numbers that
are multiple of factor 11 and belonging to one of the formulas (A) and (B). In the first interval
[3, 1157], there are 29 pairs to be removed that is computed as follows: 29=2*30-1. For all
other consecutive intervals of length equal to 1155, number of pairs to be removed is equal to
30.
Let U0 =15 be the first element of a progression whose value U0 is the number of pairs of
virtual twin prime numbers in the natural period equal to 105. The value of natural period
increases with the first "z" number of primes such that:
The number of pairs of virtual twin prime numbers in the natural period, built with first four
prime numbers,
, is equal to U1 such that
In ignoring the extra couple Cp, we get:
Let the first "z + 1" prime odd numbers identified by an index ranging from 0 to z, we denote
by Fz the zth prime number in ascending order and greater than 3. We set U0 the first term of a
progression and U0 = 15. The value of this term is equal to the number of pairs of virtual twin
prime numbers existing in the natural period equal to
with z=3.
We denote by , the number of pairs of virtual twin prime numbers present in a natural
period equal to
with z = 4. The prime number 11 is . We can write
as
follows:
In ignoring the extra couple Cp, we get:
By recurrence on "n", we get:
After simplifying:
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15 formulas that connect twin prime pairs
When the number "z" of prime numbers tends to infinity, the index "n" of the sequence Un
tends to infinity and the number of pairs of virtual twin prime numbers also tends to infinity.
We will demonstrated that:
:
The first term of the progression is obtained with z = 3, n = 0 and U0 = 15. The first natural
period
is formed with the following three prime numbers: 3, 5 and 7. The prime
numbers are taken in ascending order:
. Thus
.
The number of pairs of virtual twin numbers is infinite.
The demonstration is done.
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15 formulas that connect twin prime pairs
Conclusion
Our theory of odd numbers Np is based on the indices p of these numbers such as Np=2p+1,
. This theory leads to the proof of the theorem of the 15 formulas that connect pairs of
twin primes except the first two pairs (3, 5) and (5, 7). The reciprocal is not true. There exists
an infinity of pairs of virtual twin primes.
This approach of odd numbers provides tools to characterize the prime numbers and
composite numbers such as trigonometric functions associated with Wj and arithmetic
sequences.
Acknowledgement:
Thanks to Laurent Garnier, professor of mathematics, for helpful comments concerning this
exposition.
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15 formulas that connect twin prime pairs
References:
[1] EUCLIDE, “L œuv
’Eucl
u
l é l m p F. P y
, 1819,
tirage avec une introduction de Jean Itard, Librairie scientifique et technique Albert
Bl ch ”, P ,1993 [1,2,4].
uv u
[2] F. Mertens. J. reine angew. Math. 78 (1874), 46-62 Ein Beitrag zur analytischen
Zahlentheorie
[3] G. H. Hardy et E. M. Wright, An Introduction to the Theory of Numbers, 4e éd., p. 10.
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15 formulas that connect twin prime pairs
Appendix 1
Demonstration of the property 4-3-3-2:
a) We will determine the form of integers x1, y1 so that the pair
is written
(6q-1, 6q+1). This requirement matches the criteria of the definition 5.3.2.
b) We will also determine the mathematical form of x0 so that formulas of b1 and b2 meet the
criteria of the definition 5.3.2.
Recall:
1. Let a combination be constituted of a number of pairs of primes equal to NC with NC> 1.
Every pair of primes creates a sequence of pairs of composite number. We seek pairs of
composite numbers common to pairs of composite numbers generated by each pair of prime
numbers. Such a combination consists of two sequences of indices (p1, p2=p1+1).
2. The relations between coefficients b1 and b2 of the property 4-3-3-2 are:
- when the relation
is true with
,
the solutions are:
-
when the relation
the solutions are:
is true with
,
a) Composite numbers of the first pair of composite numbers obtained with respectively P1’’
and P2’’ are written (6q-1=x1P1’’, 6q+1= y1P2’’) with
. These integers exist
because P1’’ and P2’’ are coprimes (Property 4-2). The linear Diophantine equation
has one solution. We will determine the mathematical form of values of
x1, y1.
The odd numbers P1’’ and P2’’ are obtained by the product of distinct primes of each other
which are written as 6q-1 or 6q+1 with
. Hence, P1’’ and P2’’ are written as 6q-1 or
6q+1 with
. There are then 4 separate cases:
case-a) P1’’ and P2’’ are written respectively 6
case-b) P1’’ and P2’’ are written respectively 6
case-c) P1’’ and P2’’ are written respectively 6
case-d) P1’’ and P2’’ are written respectively 6
with
.
-1 and 6
+1 and 6
-1 and 6
+1 and 6
+1
-1
-1
+1
The mathematical form of x1 and y1 depend on each precedent case. In the first case "a", x1
P1’’ is written as 6q-1 when x1 is written as 6q1+1 with the natural integer
:
And y1 P2’’ is written as 6q+1 when y1 is written as 6q3+1 with the natural integer
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:
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15 formulas that connect twin prime pairs
The table below gives the mathematical form of x1 and y1 according to each case.
Case
a
b
c
d
P1’’
6q2-1
6q2+1
6q2-1
6q2+1
P2’’
6q4+1
6q4-1
6q4-1
6q4+1
x1
6q1+1,
6q1-1,
6q1+1,
6q1-1,
y1
6q3+1,
6q3-1,
6q3-1,
6q3+1,
(x1 P1’’)
6q-1
6q-1
6q-1
6q-1
(y1 P2’’)
6q+1
6q+1
6q+1
6q+1
The pair (x1P1’’, y1P2’’) is written as (6q-1, 6q+1). Indices of pair are (3m+1, 3m+2).
The criteria of the definition 5.3.2 are respected.
b) We will determine the mathematical form of x0 in the first formula of the system
, in
each case, so that the composite numbers with the indices b1 and b2 are written as 6q + 1
The first formula is written:
In case-a, P1’’=6q2-1 and P2’’=6q4+1, we will prove that 2*b1+3 is written as 6q+1.
Let x0 be in the following mathematical form:
with
The relation
leads to
Thus by substitution in the equation (2), one gets:
The calculation of
with the relation (2) leads to:
The pair (2*(b1-1)+3, 2*b1+3) is a pair of composite numbers that are written (6q-1, 6q+1)
multiple of the pair of odd numbers (P1’’,P2’’).
The second formula is written:
with
,
, we get:
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15 formulas that connect twin prime pairs
The table below gives the mathematical form of x0 according to each case. This form is
l
h f P1’’ w h
v
l f h
.
Case
a
b
c
d
P1’’
6q2-1
6q2+1
6q2-1
6q2+1
P2’’
6q4+1
6q4-1
6q4-1
6q4+1
x0
3t+1
3t-1
3t+1
3t-1
Examples:
Case
Formula
used
a
b
c
d
Pair (P1’’, P2’’)
P1’’
(5,13)(5,19)=(5,247)
6q2-1
(7,17)(19,17)=(133,17)
6q2+1
(5,7)(5,11)=(5,77)
6q2-1
(7,17)(7,5)(13,5)=(91,85) 6q2+1
P2’’
x0
b1
b2
6q4+1
6q4-1
6q4-1
6q4+1
3 t + 1=346
3 t - 1=11
3 t + 1= 31
3 t - 1=71
122
1928
422
5141
3581
4853
731
18062
Note: the values of b1 and b2 are related to the pairs (P1’’, P2’’) and (P2’’, P1’’). With 2
selected odd numbers, the value taken by P1’’
P2’’
h f mul
u
have not been studied in this article.
,
The second formula is factorized with the factor P1’’:
Moreover, we get:
The relation
leads to
, hence:
The pair (2*(b2-1)+3, 2*b2+3) is pair of composite numbers that is written (6q-1, 6q+1)
multiple of the pair of odd numbers (P2’’, P1’’).
The criteria of the definition 5.3.2 are respected for b1 and b2
Remarks:
If we reverse the values of P1'' and P2'': P1'=P2'', P2'=P1’', that is to say when we reverse
pairs, we obtain the following relationship:
. The formulas of b1 and b2
are then those given by the property 4-3-3-1:
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15 formulas that connect twin prime pairs
We let
and
- when the relation
solutions are:
-
such as:
is true with
when the relation
solutions are:
, the
is true with
Then we get:
, the
.
Examples:
Here are precedent examples with the property 4-3-3-2 and so
Case
Formula
used
a
b
c
d
Pair (P1’’, P2’’)
P1’’
(5,13)(5,19)=(5,247)
6q2-1
(7,17)(19,17)=(133,17)
6q2+1
(5,7)(5,11)=(5,77)
6q2-1
(7,17)(7,5)(13,5)=(91,85) 6q2+1
P2’’
x0
b1
b2
6q4+1
6q4-1
6q4-1
6q4+1
3 t + 1=346
3 t - 1=11
3 t + 1= 31
3 t - 1=71
122
1928
422
5141
3581
4853
731
18062
Indices of pairs of composite numbers formed by the combination (5, 7)(5, 11) and those
formed by the reverse combination (7. 5)(11. 5) are:
and
Here are the same examples with the property 4-3-3-1 and so
. This is an
inversion of the pairs. The values for b1 and b2 are reversed because the pairs were reversed.
The results are identical to the combination studied and its reverse combination: b1'=b2 and
b2'=b1.
Case
a
b
c
d
Formula
used
Pair (P1’, P2’)
P1’
(13,5)(19,5)=( 247,5)
6q2+1
(17,7)(17,19)=(17,133)
6q2-1
(7,5)(11,5)=(77,5)
6q2-1
(17,7)(5,7)(5,13)=(85,91) 6q2+1
P2’
x0
b1'
b2'
6q4-1
6q4+1
6q4-1
6q4+1
3 t + 1=7
3 t - 1=86
3 t - 1= 2
3 t + 1=76
3581
4853
731
18062
122
1928
422
5141
Indices of pairs of composite numbers formed by the combination (7,5)(11,5) and those
formed by the reverse combination (5,7)(5,11) are:
and
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