First Case of Fermat’s Last Theorem

Joseph Amal Nathan Reactor Physics Design Division, Bhabha Atomic Research Centre, Mumbai-400085, India email:[email protected] Abstract: In this paper two conjectures are proposed based on which we can prove the first case of Fermat’s Last Theorem(FLT) for all primes p ≡ −1(mod 6). With Pollaczek’s result [1] and the conjectures the first case of FLT can be proved for all primes greater than 3. With a computer Conjecture1 was verified to be true for primes ≤ 2437 and Conjecture2 for primes ≤ 100003. Fermat’s Last Theorem(FLT): The equation xl+ yl = zl with integral l > 2, has no solution in positive integers x, y, z. There is no loss in generality if x, y, z are pairwise prime and, if l = 4 and all odd primes only. Let p denote a prime greater than 3. The results presented here are when p ∤ xyz, known as ‘ first case of FLT ’. We will denote greatest common divisor of integers α, β, γ, ... by (α, β, γ,...). Let us define a Polynomial Fn(x, y) = (x+y) n−xn−yn for any positive odd integer n. Expanding (x+ y)n and collecting terms having same binomial coefficient we get, Fn(x, y) = xy(x+ y) 