Download This phenomenon of primitive threes of Pythagoras owes it`s

V.S.Yarosh
THE SHORTEST AND THE EASIEST
PHENOMENOLOGICAL PROOF
OF THE LAST THEEOREM OF PIERRE FERMAT
УДК 511
ББК 22.13
Р49
PACS numbers: 02.10.Ab , 02.30.Xx
In book " The last theorem of P.Fermat" by G. Edwards» ,translated from
English into Russia and published by "Mir" publishing House in 1980 in
Moscow, we read on page 14 (see also [2] ) :
«Written in Latin language article by Fermat says that
“from the other hand it's impossible to express cube in form of sum
of two cubes, or fourth degree –as a sum of two forth degrees or,any number ,
which is of higher degree then the second one, can not be written as a sum
of two such degrees. I have a surprising proof of this affirmation , but I have too
little space to describe it.”
This simple affirmation, that can be symbolically written the following way :
“for every whole number n > 2 equation x  y  z is insoluble” , is known now as the last theorem of Fermat.»
The end of the quotation. See [1] , [2].
n
n
n
PROOF
Using primitive of Pythagor triads
(a0 , b0 , c0 ) , we construct
a  n a 0
2
b  n b0
2
c  co
2
n
a solution :
(1)
Primitive equation of Fermat- Diophantus :
n
n
a  b  c
Making numbers
(a , b , c )
n
(2)
of solution (1) into even degree 2n , we write:
2n
 a0
b
2n
 b0
4
c
2n
 c0
4
a
4
(3)
Forms (3) make the solution of the equation :
a n  bn  c n
(4)
in which:
a  a
2
b  b
2
c  c
2
(5)
Analyzing forms(3) – (5) ,we come to conclusion, that
Fermat was right , as far as from solution (3) are formed equivalent
INEQUALITIES ,MADE IN FORM OF WHOLE NUMBERS :
2n
2n
a  b  c
4
2n
4
a0  b0  c0
(6)
4
(7)
Inequality (7) proves Fermat's affirmation about forth degrees of whole
numbers.
This is the case n = 4 , proved by Ferma himself. (See
[1],[2],[3] ).
Fermat also was right according to cubes of whole numbers.
Making numbers of solution (1) into degree 3n , we write:
3n
 a0  a0
b
3n
 b0  bo
4
2
c
3n
4
2
a
4
 c0  co
2
(8)
From (8) there comes inequality :
4
2
4
2
( a0  a0 )  (bo  b0 ) 
4
2
 ( c0  c0 )
(9)
Proving Fermat's affirmation according to cubes, as far as relative
form of group of numbers (9), taken from deviding
2
of all members of the form (9) into 0 :
c
4
2
2
4
2
2
[a0  (ao / c0 )]  [b0  (b0 / c0 )]  c0
4
(10)
contains multipliers:
2
2
2
2
( a 0 / c0 )  1
(b0 / c0 )  1
(11)
Which are less then unity according to known definition
primitive of Pythagor triads:
a0  v 2  u 2
b0  2vu
c0  v  u
2
in which
2
v  u natural (whole) numbers of different evenness
(12)
.
CONSEQUENCE
According to definition (8) ,affirmation of Fermat
dwells not only on cubes and squares,
but also on any variety , forming at degrees
kn , where k is any whole number (even or uneven)
and n > 2 .
Bibliography
1. Г.Эдвардс, Последняя теорема Ферма, пер. с англ.,
М., «Мир»,(1980), с.14, 22 – 28.
2. Harold M.Edwards, Fermat”s Theorem, Springer-Verlag,
New York Heidelberg Berlin (1977), s.14, 22-28.
3. М.М.Постников, Теорема Ферма, М., «Наука» , (1978), с.27-30.
APPENDIX
RECONSTRUCTION
Of ‘Really Amazing Proof’
of the Great or Last Theorem of Pier Fermat (1601 –1665).
The are enough reasons to assume that P. Fermat
considered the equation:
a n  bn  cn
(1)
as an infinite system of equations with finite number
of variable n a t u r a l numbers (a, b, c, n):
a2  b2  c2
a3  b3  c3
a4  b4  c4
.................
(2)
.................
an  bn  cn
He knew that the equation (1) at n = 2 came
from extreme antiquity. It has geometrical interpretation
and is called the equation of Pythagoras (approx. 580 – 500 B.C.):
a 2  b2  c2
(3)
It was also known that among the infinity aggregate
of solutions of equation (3) there are such threes
of numbers (a0, b0, c0) that do not have common multipliers.
These threes of
n a t u r a l numbers are known as p r i m i t i v e t h r e e s
of Pythagoras. These primitive threes of Pythagoras
were also known to be generated by
any couple v > u of natural numbers
of different parity with the help of three invariant forms:
a0  v 2  u 2
b0  2  v  u
c0  v  u
2
(4)
2
The further line of argument is obvious.
Four variable natural numbers (a0, b0, c0, n)
are building three linear forms:
a 0n  a 02  a 0n  2  (v 2  u 2 ) 2  (v 2  u 2 ) n  2
b0n  b02  b0n  2  (2  v  u ) 2  (2  v  u ) n  2
c c c
n
0
2
0
n2
0
 (v  u )  (v  u )
2
2
2
2
2
(5)
n2
forming the equation, which makes sense only at n = 2:
a02  a0n2  b02  b0n2  c02  c0n2
(6)
For all other n > 2 form (6) makes sense of inequality
a02  a0n 2  b02  b0n 2  c02  c0n 2
(7)
Inequality (7) can be written down in the following for
a 02  (
a0 n2
b
)  b02  ( 0 ) n 2  c02
c0
c0
(8)
Inequality (8) is turning into equality
with the help of corrective multiplier φn-2:
a02  (
a0 n 2
b
)  b02  ( 0 ) n 2  c02   n 2
c0
c0
(9)
An i n v a r i a n t form of multiplier φn-2 follows from equality (9)
and is computable for
any couples v > u
of natural numbers of different parity:
a 02  (
 n2 
a0 n2
b
)
 b02  ( 0 ) n  2
c0
c0
c 02
(10)
As a result we acquire equivalent of equation (1)
computable for all n ≥ 2:
a*n  b*n  c*n
(11)
in which:
a*n  a 02  (
b*n  b02  (
a0 n2
)
c0
b0 n  2
)
c0
(12)
c*n  c 02   n  2
Formulas for calculation of primitive threes (a*, b*, c*)
of Fermat – Deophant follow from (12)
a*  n a 02  (
b*  n b02  (
a0 n2
)
c0
b0 n  2
)
c0
c*  n c 02   n  2
Entry into the infinity aggregate
of not primitive threes is carried out by the means
of multiplication of primitive threes by any common multiplier:
(13)
a  a*  S
b  b*  S
(14)
c  c*  S
Thus according to (4) the following correlations always exist:
(
a0
 1)
c0
and
(
b0
 1)
c0
(15)
by virtue of which:
FOR ALL n > 2 MULTIPLIER OF PROPORTIONALITY ,
CALCULATED WITH THE HELP OF INVARIANT FORM (10),
PRIMITIVE THREES OF FERMAT-DEOPHANT,
CALCULATED WITH THE HELP OF INVARIANT FORMS (13),
CANNOT BE WHOLE NUMBERS.
CONCLUSION
THE RECORD MADE BY PIER FERMAT
ON NERROW MARGINE OF ARITHMETIC OF DEOPHANT
IS CORRECT. THERE CAN BE
NO DOUBT ABOUT IT DESPITE THE OPPINION
PREVALENT IN WORKS OF THEORY OF NUMBER SPECIALISTS.
In summary I’d like to point out the following.
My work ‘About some mistaken statement in theory of
number
and completeness of the final solution of Fermat’s theorem’ is published
in collection of scientific works of the State University of Tula
in 1995 on pages 130-137.
In this work attention of readers is drawn
to the following property of formulas for calculation
infinity aggregates of primitive threes (a*, b*, c*) of Fermat – Deophant:
ALL FORMULAS CONTAIN SQUARES
2
2
2
OF PRIMITIVE THREES OF PYTHAGORAS ( a 0 , b0 , c 0 ).
AND REGARDLESS OF THE CONSTANT
OF PROPORTIONALITY THESE SQUARES ARE MULTIPIED BY,
ROOTS WITH DEGREES n > 2 WILL ALWAYS BE IRRATIONAL.
The numbers φn-2 generated by the form (10).
Could be used as constants of proportionality.
But there could be any natural numbers N = 1, 2, 3, 4. …
But there may be also an appropriate combination
of primitive threes of Pythagoras:
1
Dn   (a 0n  2  b0n  2  c 0n  2 )
3
(16)
described in the book ‘Finale of the Centuries-old Enigma
of Deophant and Fermat’.
This phenomenon of primitive threes of Pythagoras owes it’s
existence to two properties of numbers (a0, b0, c0):
1. Threes (a0, b0, c0) don’t have common multipliers
2. Each of these numbers may be factorized
into prime factors only in one way.
V.S.Yarosh