Fermat and Pythagoras Divisors for a New Explicit Proof of Fermat’s Theorem:a4 + b4 = c4. Part I

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: (F0): a14 + (2sb1)4 = c14, is not solvable in N, (where a1,b1,c1 ∈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 a2,b2,γ2 ∈2N +1, such that (F1): a24 + (2s-1β2)4 = γ24, 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 (F0) , then (a2, 2s-1β2, γ2) is also a solution of Fermat’s type, but with 2 ≤ s-1< s, witch is absurd. To reach such a result, we suppose first that (F0) is solvable in (a1,2sb1,c1) , s ≥ 2 like above; afterwards, proceeding with “Pythagorician divisors”, we creat the notions of “Fermat’s b-absolute divisors”: (db,d′b) which it uses hereafter. Then to conclude our proof, we establish the following main theorem: there is an equivalence...