TITLE:
Fermat and Pythagoras Divisors for a New Explicit Proof of Fermat’s Theorem:a4 + b4 = c4. Part I
AUTHORS:
Prosper Kouadio Kimou, François Emmanuel Tanoé, Kouassi Vincent Kouakou
KEYWORDS:
Factorisation in ℤ, Greatest Common Divisor, Pythagoras Equation, Pythagorician Triplets, Fermat's Equations, Pythagorician Divisors, Fermat's Divisors, Diophantine Equations of Degree 2, 4-Integral Closure of ℤ in ℚ
JOURNAL NAME:
Advances in Pure Mathematics,
Vol.14 No.4,
April
29,
2024
ABSTRACT: In this paper we prove in a new way, the well known result, that Fermat’s equation a4 + b4 = c4, is not solvable in ℕ , when abc≠0 . To show this result, it suffices to prove that: ( F 0 ): a 1 4 + ( 2 s b 1 ) 4 = c 1 4 , is not solvable in ℕ , (where a 1 , b 1 , c 1 ∈2ℕ+1 , pairwise primes, with necessarly 2≤s∈ℕ ). The key idea of our proof is to show that if (F0) holds, then there exist α 2 , β 2 , γ 2 ∈2ℕ+1 , such that ( F 1 ): α 2 4 + ( 2 s−1 β 2 ) 4 = γ 2 4 , holds too. From where, one conclude that it is not possible, because if we choose the quantity 2 ≤ s, as minimal in value among all the solutions of ( F 0 ) , then ( α 2 ,2 s−1 β 2 , γ 2 ) is also a solution of Fermat’s type, but with 2≤s−1