A new generalization of Fermat's Last Theorem

Cai, Tianxin; Chen, Deyi; Zhang, Yong

doi:10.1016/j.jnt.2014.09.014

Cited by 3 publications

(2 citation statements)

References 9 publications

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“…The right sides of equalities ( 12) and ( 13) are equal integers. However, the number on the right side of equality (12) has a divisor   n m − ( ), and the number on the right side of equality (13) does not have such a divisor.…”

Section: Mathematicsmentioning

confidence: 99%

The proof of Fermat’s last theorem based on the geometric principle

Gevorkyan

2023

Eureka: PE

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This paper provides another proof of Fermat's theorem. As in the previous work, a geometric approach is used, namely: instead of integers a, b, c, a triangle with side lengths a, b, c is considered. To preserve the completeness of the proof of the theorem in this work, the proof is repeated for the cases of right and obtuse triangles. In this case, the Fermat equation ap+bp=cp has no solutions for any natural number p>2 and arbitrary numbers a, b, c. When considering the case when the numbers a, b, c are sides of an acute triangle, it is proven that Fermat’s equation has no solutions for any natural number p>2 and non-zero integer numbers a, b, c. Numbers a=k, b=k+m, c=k+n, where k, m, n are natural numbers that satisfy the inequalities n>m, n<k+m, exhaust all possible variants of natural numbers a, b, c, which are the sides of the triangle. In an acute triangle, the following condition is additionally satisfied: To study the Fermat equation, an auxiliary function f(k,p)=kp+(k+m)p–(k+n)p, is introduced, which is a polynomial of natural degree p in the variable k. The equation f(k,p)=0 has a single positive root for any natural A recurrent formula connecting the functions f(k,p+1) and f(k,p) has been proven: f(k,p+1)=kf(k,p)-[n(k+n)p-m(k+m)p]. The proof of the main proposition 2 is based on considering all possible relationships between the assumed integer solution of the equation f(k,p+1)=0 and the number corresponding to this solution The proof was carried out using the mathematical apparatus of number theory, 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 theorem were proved

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Section: Mathematicsmentioning

confidence: 99%

The proof of Fermat’s last theorem based on the geometric principle

Gevorkyan

2023

Eureka: PE

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“…Willes' proof is quite complex to understand and may not have been known in Fermat's time. After Willes' proof, other researchers have also provided an alternative to the Fermat Last Theorem through several new Diophantine and Hilbert-Waring hybrid equations [2]. In [3], Fermat's Last Theorem is proved by using Reduction ad absurdum, the Pythagorean Theorem, and congruent triangles' properties.…”

Section: Introductionmentioning

confidence: 99%

Proving The Fermat Last Theorem for Case q≤n

Prayanti

Maxrizal²

2022

EMJ

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Fermat's Last Theorem is a well-known classical theorem in mathematics. Andrew Willes has proven this theorem using the modular elliptic curve. However, the proposed proof is difficult for mathematicians and researchers to understand. For this reason, in this study, we provide evidence of several properties of Fermat's Last Theorem with a simple concept. We use Newton's Binomial Theorem, well-known in Fermat's time. In this study, we prove Fermat's Last Theorem for case . We also use the Newton’s Binomial theorem to verify several cases .

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A variety of Euler's sum of powers conjecture

Cai¹,

Zhang²

2021

Czech.Math.J.

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A new generalization of Fermat's Last Theorem

Cited by 3 publications

References 9 publications

The proof of Fermat’s last theorem based on the geometric principle

The proof of Fermat’s last theorem based on the geometric principle

Proving The Fermat Last Theorem for Case q≤n

A variety of Euler's sum of powers conjecture

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