# Geometric approach to the proof of Fermat’s last theorem

### Abstract

A geometric approach to the proof of Fermat’s last theorem is proposed. Instead of integers *a*, *b*, *c*, Fermat’s last theorem considers a triangle with side lengths *a*, *b*, *c*. It is proved that in the case of right-angled and obtuse-angled triangles Fermat's equation has no solutions. When considering the case when *a*, *b*, *c* are sides of an acute triangle, it is proved that Fermat's equation has no entire solutions for *p*>2. The numbers *a*=*k*,* b*=*k*+*m*,* c*=*k*+*n*, where *k*, *m*, *n* are natural numbers satisfying the inequalities *n*>*m*, *n*<*k*+*m*, exhaust all possible variants of natural numbers *a*, *b*, *c* which are the sides of the triangle.

The proof in this case is carried out by introducing a new auxiliary function *f*(*k*,*p*)=*k ^{p}*+(

*k*+

*m*)

*–(*

^{p}*k*+

*n*)

*of two variables, which is a polynomial of degree in the variable . The study of this function necessary for the proof of the theorem is carried out. A special case of Fermat’s last theorem is proved, when the variables*

^{p}*a*,

*b*,

*c*take consecutive integer values. The proof of Fermat’s last theorem was carried out in two stages. Namely, all possible values of natural numbers

*k*,

*m*,

*n*,

*p*were considered, satisfying the following conditions: firstly, the number (

*n*–

^{p}*m*) is odd, and secondly, this number is even, where the number (

^{p}*n*–

^{p}*m*) is a free member of the function

^{p}*f*(

*k*,

*p*).

Another proof of Fermat’s last theorem is proposed, in which all possible relationships between the supposed integer solution of the equation *f*(*k*, *p*)=0 and the number corresponding to this supposed solution are considered.

The proof is carried out using the mathematical apparatus of the theory of integers, elements of higher algebra and the foundations of mathematical analysis. These studies are a continuation of the author's works, in which some special cases of Fermat’s last theorem are proved

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### References

Tanchuk, M. (2016). Rozghadka taiemnytsi dovedennia velykoi teoremy Piera de Ferma. Trysektsiya dovilnykh ploskykh kutiv i kvadratura kruha. Kyiv: DETUT, 34.

Cox, D. A. (1994). Introduction to Fermat's Last Theorem. The American Mathematical Monthly, 101 (1), 3–14. doi: https://doi.org/10.2307/2325116

Kleiner, I. (2000). From Fermat to Wiles: Fermat's Last Theorem Becomes a Theorem. Elemente der Mathematik, 55 (1), 19–37. doi: https://doi.org/10.1007/pl00000079

Postnikov, M. M. (1978). Teorema Ferma. Vvedenie v teoriyu algebraicheskikh chisel. Moscow: Nauka, 128. Available at: https://ikfia.ysn.ru/wp-content/uploads/2018/01/Postnikov1978ru.pdf

Edvards, G. (1980). Poslednyaya teorema Ferma. Geneticheskoe vvedenie v algebraicheskuyu teoriyu chisel. Moscow: Mir, 486. Available at: http://www.ega-math.narod.ru/Books/Edwards.htm

Wiles, A. (1995). Modular elliptic curves and Fermat’s last theorem. The Annals of Mathematics, 141 (3), 443. doi: https://doi.org/10.2307/2118559

Taylor, R., Wiles, A. (1995). Ring-Theoretic Properties of Certain Hecke Algebras. The Annals of Mathematics, 141 (3), 553. doi: https://doi.org/10.2307/2118560

Leng, S. (1979). Vvedenie v teoriyu modulyarnykh form. Moscow: Mir, 256. Available at: https://ua1lib.org/book/440534/beb856?id=440534&secret=beb856

Agafontsev, V. V. (2012). Velikaya teorema Ferma (neobychniy podkhod). Materialy X Mezhdunar. zaoch. nauch.-prakt. konf. «Innovatsii v nauke». Ch. 1. Novosibirsk: Izd-vo Sibirskaya assotsiatsiya konsul'tantov, 6–10. Available at: https://cyberleninka.ru/article/n/velikaya-teorema-ferma-neobychnyy-podhod

Andronov, I. K. (1975). Matematika deystvitel'nykh i kompleksnykh chisel. Moscow: Prosveschenie, 158. Available at: http://publ.lib.ru/ARCHIVES/A/ANDRONOV_Ivan_Koz%27mich/_Andronov_I.K..html#0005

Kurosh, A. G. (1968). Kurs vysshey algebry. Moscow: Nauka, 431. Available at: http://ijevanlib.ysu.am/wp-content/uploads/2018/03/Kurosh1968ru.pdf

Gevorkyan, Yu. L. (2020). Chastniy sluchay teoremy Ferma. Trudy IX Mezhdunarodnoy nauchno-prakticheskoy konferentsii “Scientific achievements of modern society”. Liverpul', 418–429.

Gevorkyan, Y. L. (2020). Fermat’s Theorem. Scientific Journal of Italia "Annali D’Italia”, 1 (8), 7–16. Available at: https://www.calameo.com/books/006103417d8de062343b8

*EUREKA: Physics and Engineering*, (4), 127-136. https://doi.org/10.21303/2461-4262.2022.002488

Copyright (c) 2022 Yuriy Gevorkyan

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