This text represents my work on the solutions of the Diophantine Equation: αν = βν + γν, for ν = 2, 4, 3, 5, 7, 11… (odd prime). Recognizing that this difficult work has many errors, I will appreciate it very much, if in time, you send to me via the publisher, your comments and corrections, so that the next edition be better. The proof of the non-existence of solutions of the above equation, for: ν = 3, 4, 5, 7, 11…, will be and proof of the relative FERMAT’S proposal with title: “FERMAT’S LAST THEOREM” (F.L.T.).