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Fermat's Last Theorem, Solution Sets v6

Ernest R. Lucier
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Abstract

The non-zero integer solution set is derived for C^n = A^n + B^n. The non-zero integer solution set for n = 2 is [C - (a + b)]^2 = 2ab. The variables a and b equal (C - A) and (C - B) respectively and are nonzero integer factors of 2M^2 where M is a non-zero integer. C is greater than (a + b) since the square root of 2ab is an imaginary number when C is less than (a + b). C is equal or greater than (a + b + 1) since we are only considering whole numbers. The derivation of the solution set for n = 2 is applied to n = 3, n = 4, and generalized to n. The solution set for n = n is [C - (a + b)]^n = ab([n:2]C^(n-2)*(2) - [n:3]C^(n-3)*(3a + 3b)+ ... + [n:n] {[n:1]a^(n-2) + [n:2]a^(n-3)*b^1 + [n:3]a^(n-4)*b^2 +... + [n:n-1]b^(n-2)}). Where the binomial coefficient [n:r] = n!/[(n - r)!r!] is the coefficient of the x^r term in the polynomial expansion of the binomial power (1 + x)^n and [n:r] = 0 if r > n. Divide this equation by [C-(a + b)]^(n-2) to obtain [C-(a + b)]^2. The solution set for [C - (a + b)]^2 equals 2ab. The nth solution set equals 2ab only when n equals 2. [C - (a + b)]^2 (I.e., [C - (a + b)]^n divided by [C - (a + b)]^(n-2)) is always greater than 2ab when n is greater than 2. Non-zero integer solutions exist only for n = 2.

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Ernest R. Lucier. 2012-02-24. Fermat's Last Theorem, Solution Sets v6. https://arxiv.org/abs/math/0502245

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