Number Theory

    

Fermat's Last Theorem: A Proof by Contradiction

Authors: Benson Schaeffer

In this paper I oer an algebraic proof by contradiction of Fermat's Last Theorem. Using an alternative to the standard binomial expansion, (a+b)n = an + b Pn i=1 ani(a + b)i1, a and b nonzero integers, n a positive integer, I showthat a simple rewrite of the equation stating the theorem, Ap + (A + b)p = (2A + b c)p; A; b and c positive integers, entails the contradiction of two positive integers that sum to less than zero,(2f + g)(f + g)(f + g + b) Xp2 i=1 (2f + g)p2i(3f + 2g + b)i1 + (f + b)(f + g)(3f + 2g + b)p2 + fb(3f + 2g + b)p2 < 0; f and g positive integers. This contradiction shows that the rewrite has nonon-trivial positive integer solutions and proves Fermat's Last Theorem.

Comments: 6 Pages.

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[v1] 2022-08-06 01:37:24

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