Download NUMBER FRACTION AND RELATED FUNCTIONS

NUMBER FRACTION AND RELATED FUNCTIONS
In several earlier notes we have discussed the important new function termed the
Number Fraction and defined as-
f (N ) 
 ( N )  ( N  1)
N
Here σ(N) is the sigma function( also known as the divisor function) representing the sum
of all divisors of N. Thus N=36 has σ(36)= 91 and f(36)=3/2. What is important about the
f(N) function is that it vanishes whenever N is a prime and seems to be bounded to less
than about 5 for all N (This conjecture has not yet been proven and it may turn out that
still larger values of f(N) exist). So far the largest f(N) we have been able to find is
f(15805424194560000)= 4.269212092… . Note here that N+1=15805424194560001 is a
prime so that f(N+1)=0.
We term those numbers with f(N)>1 super-composites , those with 0<f(N)<<1 as
semi-primes, and those for which f(N)=0 as primes. Examples of these three groups are
f(7524)= 1.902.. , f(3007)= 0.04256.. , and f(4723)=0. We want here to discuss some
additional properties of the f(N) function and to introduce some related functions F(N)
and G)N) which directly indicate the location of prime numbers.
We begin( as already done in an earlier note of Sept. 1, 2012) with the following line plot
of f(N) versus N in the range 1<N<86-
One clearly sees from the graph that the largest values of f(N) occur when N is a multiple
of six(ie-N=6n) and f(N)=0 at the locations of what we call the Q primes. The Q primes
all have the form 6n±1 and yield essentially the entire set of primes in any given range
above N=3. This fact is supported by the hexagonal integer spiral shown which locates all
Q primes along the diagonals 6n±1 removed by π/3 radians from the 6n line-
If we look at f(pn), where p is a prime number, we findf ( p)  0
f ( p 2 )  [1] / p )
f ( p 3 )  [1  p )] /( p 2 )
f ( p 4 )  [1  p  p 2 ] / p 3
This allows one to conclude that the number fraction for the nth power of a prime is justf ( pn ) 
1
p n 1
n 2
 pk 
k 0
(1  p1n )
( p  1)
after summing the finite geometric series indicated. Thus we have, for example , that
1  114  1464
f (1610510)  f (115 )  

 10  14641
An interesting result follows from substituting the above form for f(pn)into the quotient-
[ f ( p n )  1] p (1  p  n )
p

(1  p n )
f ( p n 1 )
On setting n=2, this introduces the new functionF(N ) 
f (N 2 )  1
f (N 3 )
which yields prime numbers only when N=p. For other integers N the function has values
lying below F(N)=N. A very clear picture of this is obtained by plotting F(N) versus N as
shown-
We see here that F(N)equals N only when N is a prime N=p. We will call F(N) the Prime
Number Function since its local maxima always indicate primes. For N≠p the function
F(N) has values less than N but never vanishes. As N gets larger the spacing between the
primes increases in accordance with the prime number theorem. If we look in the
immediate vicinity of the Mersenne Prime at N=217-1=131071, one gets the following
picture-
More than anything else this shows the futility of trying to find more than the presently
known 48 Mersenne primes, since it indicated there are an infinite number of other
primes of the formN=kn –m
Where k and n are specified and m is varied until a prime is found. These are just as
legitimate as any Mersenne Prime. One could keep up the calculations forever without
finding even a small fraction of the existing primes. For example, we have the primes217-9=131063 ,
317-10=129140153,
517-16=762939453109 ,
and
247+5=140737488355333.
Another consequence of the above Prime Number Formula F(N) is that it is essentially
equivalent to a criterion for a number being prime. Thus we have for example1
(  1)
[ f (121)  1]
F (11) 
 11
 11
12
f (1331)
121
showing N=11 to be a prime. If we try N=9 we find the ratio will equal
[f(81)+1]/f(729)=360/121= 2.975.. and hence not 9. Thus N=9 is a composite number.
It remains to be answered what is the value of f(N) when N is composite. We already
know that when N is a power of a prime p (meaning also that N is odd) we can write
down at once the value of f(N). This however leaves all even numbers with the exception
of powers of two and many odd numbers not expressible in integer powers of p. Under
those conditions one must use brute force to find f(N). Certainly N=129 is not a power of
p while 128 is. For these two cases we have[3  43] 46
f (129) 

129
129
(1  2 6 ) 63
f (128)  f ( 2 ) 

( 2  1)
64
7
and
We also have that f(127)=0 since 127 is a prime. The fact that any number N can be
expressed as the product of primes taken to specified powers is of little help here since
generally f(aˑb)≠f(a)ˑf(b) and f(a+b)≠f(a)+f(b).
Another function related to both f(N) and F(N) isG( N ) 
[ ( N 2 )  1]
F(N )
,

[ ( N 3 )  1  N 3 ]
N
N 2
which represents essentially a normalized version of F(N) with the property that G(p)=1
for all primes p. Here are a few examples of G(p)G(1103)=1, G(287137)=1, and G(173467034537623)=1
What is convenient about this form of representing primes is that σ(N) is a built in
function in many computer mathematics programs such as MAPLE and
MATHEMATICA so that G(N) is easily found. The criterion for prime and composite
numbers becomesA number N is prime if and only if G(N)=[σ(N2)-1]/[σ(N3)-1-N3 ] =1. Otherwise it is a
composite.
A bit of manipulation allows us to write this last criterion in the compact form-
N
1
f (N 3 )  f (N 2 )
So if we ask what type of number N=862367891772543608539977119114066279 is,
the answer is it is prime since G(N)=1. The number N=59827441207983573578215783
is composite since G(N)=1.169... x 10-25. The Fermat number N=232+1=4294967297 has
G(N)=1.492..x 10 -7 and so is composite. The Mersenne Number M=231-1=2147483647
is prime since G(M)=1. Of course one can always go back to the basic definition of f(N)
and from its evaluation show that a number is prime if f(N)=0 and composite if f(N)≠0.
Thus f(216+1)=f(65537)=0 must be prime.
October 2013