Results Beyond Fermat's Last Theorem International Journal of Mathematics Trends and Technology (IJMTT) © 2022 by IJMTT Journal Volume-68 Issue-8 Year of Publication : 2022 Authors : Jagjit Singh Patyal 10.14445/22315373/IJMTT-V68I8P511

How to Cite?

Jagjit Singh Patyal, "Results Beyond Fermat's Last Theorem," International Journal of Mathematics Trends and Technology, vol. 68, no. 8, pp. 116-128, 2022. Crossref, https://doi.org/10.14445/22315373/IJMTT-V68I8P511

Abstract
In this paper, some results relating to Fermat's last theorem and beyond this theorem, have been presented. The expression of the form (x+y)n-(x-y)n, where x,y are variable positive integers and x>y, has been analyzed to derive some results relating to the Diophantine equation an=an1+an2+...+ans, where a,a1,a2,..., as are positive integers. An attempt has been made to give a simple proof of Fermat's last theorem and further this theorem has been extended to the case of s=3 relative to the equation an=an1+an2+...+ans. A result as a theorem 2.1 has been given to find the least positive integral value of s in the equation an=an1+an2+...+ans. A solution of each of the equations a2=a21+a22+...+a2n and a3=a31+a32+a33+a34 has been obtained. It has been proved that the equation an=an1+an2+...+ans can be expressed as (u+v)n-(u-v)n=(2a2)n+...+(2as)n, where u+v=2a, u+v=2a1. It will also be shown that the Diophantine equation an=an1+an2+...+ans is a particular case of the equation (x+y)n = (x-y)n+2(n1)xn-1y+2(n3)xn-3y3+...+2α, α={ yn,(nn-1)xyn-1, if n is odd if n is even as it is obtained by putting some positive integeral values u,v(u>v) of x,y respectively. Finally equation an=an1+an2+...+ans has been analyzed to conclude this paper.

Keywords : Diophantine equation, expression, function, number of terms, positive integer.

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