On Direct Proofs of Fermat’s Last Theorem: Abel Conjecture, the Even and Non-Prime Exponents

On Direct Proofs of Fermat’s Last Theorem: Abel Conjecture, the Even and Non-Prime Exponents, and the First Case
Kimou Kouadio Prosper (Institut Polytechnique Félix Houphouët-Boigny, Côte d’Ivoire) and Kouassi Vincent Kouakou (Université Nangui Abrogoa, Côte d’Ivoire)
Abstract
In this paper, we study Fermat's equation, with positive integers such that . Consider the set of hypothetical solutions of equation (1) and . Let be a prime, we establish the following results: This completes the direct proof of Abel's conjecture. This completes the direct proof of the second case of even exponent FLT. if is a non-prime odd integer. If then . This provides simultaneous Diophantine evidence for the first case of FLT and the second case . We analyse each of the evidence from the previous results and propose a ranking in order of increasing difficulty to establish them.
Keywords
Fermat Last Theorem, Fermat equation, First case, Secund case, Abel Conjecture, Kimou main divisors Theorem. The even exponent, The odd non-prime exponent, a prime number
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