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ON THE ISSUE OF THE RELATIONSHIP BETWEEN FERMAT'S LAST THEOREM,

FLT, AND THE PYTHAGOREAN THEOREM.

Sergey P. Klykov1,2

Fermentation expert, Poste Restante, Obolensk ., Serpukhov district, Moscow region, Russia,

142279.

1Corresponding author.

2Alpha-Integrum, Ltd.

Email addresses: smlk03@mail.ru

ABSTRACT.

This article shows that the FLT equation, an+bn =cn and the equation derived from it, (bc)-n+(ac)-n =(ab)-n, are

showing different numbers of possible rational and irrational solutions. Taking into account the proof of the

Pythagorean theorem and the previously published symmetry for the solutions of the Pythagorean theorem, the

shortest solution for the Fermat's last theorem, FLT, is shown. It was based on point of view, that the FLT is the

same thing as the Pythagorean theorem by the way: an+bn =cn→ a(n/2)2+b(n/2)2 =c(n/2)2→A2+B2=C2.

Key words: Fermat's last theorem (FLT); Pythagorean theorem; Fermat's last theorem proof.

INTRODUCTION.

THEOREM: Fermat’s last theorem (FLT), has been proven here [1], and reads: No three

positive integers a, b, and c satisfy the equation an + bn = cn for any integer value of n greater

than 2.

The article [2] shows the symmetry in the ratio of rational solutions of the Pythagorean theorem

for n and -n and gives the calculation of hypothetical solutions for FLT and similar conditions

(see below), which do not give such symmetry due to obtaining complex numbers in possible

solutions for n less than -2. This provides for the conditions of the FLT and similar conditions a

deficiency concerning their fullness for any considerations. It is also shown, that the number 1

or 0 can be reproduced only in paradoxical conditions that cannot correspond to reality, while the

indicator n is fundamental. It is argued that the Pythagorean theorem itself is a proof for FLT, if

we take into account the above. However the meanings of all a, b, c-values were considered as

integers a priori, in contrast to the FLT conditions, where it is not postulated. So, the n-values

were considered more preferable, if comparing with a, b, c values and the article [2] contains the

results of modeling using terms of n.

PROOF: Indeed, an elementary scheme for different n and -n values is presented to show

symmetry and asymmetry between possible rational and irrational solutions in various equations

with degrees n and –n:

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an+bn =cn→(1/a)-n+(1/b)-n=(1/c)-n→(abc/a)-n+(abc/b)-n=(abc/c)-n→(bc)-n+(ac)-n=(ab)-n, (1).

Based on Scheme (1), it is easy to determine that only for n = 2 and n = -2 is there symmetry in

the number of rational solutions for the first and last equations of this Scheme (1), i.e. all

solutions are rational.

If there is at least one irrational solution in the first equation for n greater than 2, then the last

equation of Scheme (1) gives 2 irrational solutions. Two irrational solutions of the first equation

give no more 3 irrational solutions in the last equation. That is, for n greater than 2, there is an

asymmetry in the number of rational and irrational solutions.

If consider Scheme (2):

an+bn =cn→ a(n/2)2+b(n/2)2 =c(n/2)2→A2+B2=C2, where A=an/2, B=bn/2, C=cn/2, (2).

Only 2 options are possible - when A, B, C are only rational numbers as Pythagorean triples and

when the triples A, B, C cannot simultaneously contain all 3 rational terms, but obey the laws of

the Pythagorean theorem and there is no third area for the considering within the framework of

17th century algebra:

1) Obviously, for n=2, the conditions for the Pythagorean triples in the Pythagorean theorem

are provided.

2) If n ≥3, the conditions for the Pythagorean theorem will also be provided, but for non-

integers A, or B, or C, or in pairs, or for all three numbers at the same time.

Thus, the second and subsequent equations of Scheme (2) fill the gap that was indicated in the

article [2] regarding the FLT conditions as insufficient.

CONCLUSION.

1) The conditions of symmetry and asymmetry for the distribution of rational solutions in

various equations of Pythagoras and FLT are considered.

2) The conditions are shown that make up for an insufficiency of the FLT conditions, as a variant

of true Pythagorean equation in its essence.

3) One of the conclusions of article (2) is confirmed that the Pythagorean theorem can be

evidence in favor of the truth of FLT.

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REFERENCES:

1. Wiles, Andrew. Modular elliptic curves and Fermat’s last theorem// Annals of

Mathematics : journal. — 1995. — Vol. 141, no. 3. — P. 443—551.

2. Klykov Sergey P. The problem with representing 1 as a number under the conditions of

Fermat's last theorem, FLT//Preprint. DOI: 10.13140/RG.2.2.28739.43046;

https://www.researchgate.net/publication/347999295_The_problem_with_representing_1

_as_a_number_under_the_conditions_of_Fermat's_last_theorem_FLT

DOI: 10.13140/RG.2.2.36466.02243

March, 31,2021.