Two criteria in Fermat infinite descent method - Grinberg, Anatoly A. (Open Science Journal, 2017)
A method infinite descent is traditionally used to proof the Fermat’s theorem for the special case of exponent n=4. At each step, the method sequentially generates a new Fermat’s equation with one of the term being smaller than that in the preceding step. After a finite number of steps the term becomes less than one and this is taken as criterion of the insolvency of the original Fermat’s equation. We show that the power of factor 2, in even parameter of Pythagoras’ equation solution used in the proof, decreases by one at each step of the descent. As a result we arrive at an unsolvable equation. This is the second criterion for the descent method. Which of the two criteria is reached first depends on the parameters of the initial Pythagorean solutions chosen for the analysis.