Fermat's Last Theorem, Solution Sets v6
Abstract
The non-zero integer solution set is derived for Cn = An + Bn. 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 2M2 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)*b1 + [n:3]a(n-4)*b2 +... + [n:n-1]b(n-2)). Where the binomial coefficient [n:r] = n!/[(n - r)!r!] is the coefficient of the xr 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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