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The Wonders of Arithmetic from Pierre Simon de Fermat
Год написания книги
2021
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(x+y) = a
+b
(2)
In this case, it is easily to see that there is only one way to get solutions to equation (1) when the numbers c, x, y, and x+y are cubes, i.e.
с = x+y = p
+q
= z
; x = p
; y = q
(3)
Then equation (1) must have the form:
(z
)
= (z
)
(p
+q
) (4)
Thus, we found that if there are numbers a, b, and c that satisfy equation (1), then there must be numbers p<a, q<b, and z<c that satisfy equations (3)
p
+q
= z
If we now apply the same approach to solving this equation, that we applied to solving equation (1), we will get the same equation, only with smaller numbers. However, since it is impossible to infinitely reduce natural numbers, it follows that equation (1) has no solutions in integers.
At first glance, we have received a very simple and quite convincing proof of the Fermat problem by the descent method, which no one has been able to obtain in such a simple way for 385 years, and we can only be happy about it. However, such a conclusion would be too hasty, since this proof is actually incorrect and can be refuted in the most unexpected way.
However, this refutation is so surprising that we will not disclose it here, because it opens the way not only for the simplest proof of the FLT, but also automatically allows to reduce it to a very simple proof of the Beal conjecture. The disclosure the method of refuting this proof would cause a real commotion in the scientific world, therefore we will include this mystery among our riddles (see Appendix V Pt. 41).
So, we have demonstrated here solving to Fermat's problems (only by descent method!):
1) The proof of the Basic theorem of arithmetic.
2) The proof of the Fermat's theorem on the unique solving the
equation p
= q
+ 2.
3) A way to prove Fermat's Golden Theorem.
4) A Fermat's way to solve the Archimedes-Fermat equation
Ax
+ 1 = y
.
5) The proof method of impossibility a
+b
=c
in integers, which
opens a way to simplest proofs of the FLT and Beal conjecture.
6) A Fermat's proof his grandiose discovery about primes in the
form 4n + 1 = a
+ b
which we have presented in another style in
Appendix IV, story Year 1680.
Over the past 350 (!!!) years after the publication of these problems by Fermat, whole existing science could not even dream of such a result!
3.5. Parity Method
Before we embarking on the topic "Fermat's Last Theorem" we note that this problem was not solved by Fermat himself using the descent method, otherwise in his FLT formulation there would be no mention of a "truly amazing proof", which certainly related to other methods. Therefore, to the above examples of the application of the descent method we will add our presentation of two methods unknown to today's science. The most curious of these is the parity method.
3.5.1. Defining Parity as a Number
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