On a Diophantine Proofs of FLT: The First Case and the Secund Case z≡0 (mod p) and Significant Related Problems

In this paper, we study Fermat's equation, 𝑥 𝑛 + 𝑦 𝑛 = 𝑧 𝑛 (1) with 𝑛 > 2, 𝑥, 𝑦, 𝑧 non-zero positive integers. Let (𝑎, 𝑏, 𝑐) be a triple of non-zero positive integers relativity prime. Consider the equation (1) with prime exponent 𝑝 > 2. We establish the following results: - 𝑎 𝑝 + 𝑏 𝑝 ≠ (𝑏 + 1) 𝑝 . This completes the general direct proof of Abel's conjecture only prove in the first case 𝑎𝑏(𝑏 + 1) ≢ 0 (𝑚𝑜𝑑 𝑝). - 𝑎 2𝑝 + 𝑏 2𝑝 ≠ 𝑐 2𝑝 . This completes the direct proof of Terjanian Theorem only prove in the first case 𝑎𝑏𝑐 ≢ 0 (𝑚𝑜𝑑 𝑝)). - 𝑎 𝑛 + 𝑏 𝑛 ≠ 𝑐 𝑛with𝑛 is a non-prime integer.A new result almost absent in the literature of this problem. - If 𝑎𝑏 ≢ 0 (𝑚𝑜𝑑 𝑝) then𝑎 𝑝 + 𝑏 𝑝 ≠ 𝑐 𝑝 . This provides simultaneous Diophantine evidence for the first case oand the second case𝑐 ≡ 0 (𝑚𝑜𝑑 𝑝) of FLT. We analyse each of the evidence from the previous results and propose a ranking in order of increasing difficulty to establish them.