Kimou Kouadio Prosper¹ and Kouassi Vincent Kouakou², ¹Institut Polytechnique Félix Houphouët-Boigny, Côte d’Ivoire, ²Université Nangui Abrogoa, Côte d’Ivoire

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.

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