Abstract
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.
Highlights
It is convenient to write Fermat’s equation with exponent n= 4 in the form (I.1)The variables X, Y, Z are positive, pair-wise relatively prime integers, where X is even, and Y, Z are odd [1-8].Open Science Journal – December 2017Open Science Journal Research ArticleEquation (I.1) shows that the squares of these variables,, must coincide with the uniquely defined solutions of Pythagoras’ equation, which are determined by the following expressions (I.2)where s and u being the odd and the even positive parameters, respectively [4], subject to inequality s > u and (s, u)=1
After some finite number of steps the term becomes less than one and this is taken as the criterion of insolvency of Fermat’ equation in integer numbers
After some number of steps the parameter becomes odd and we arrive at an unsolvable equation
Summary
It is convenient to write Fermat’s equation with exponent n= 4 in the form (I.1)The variables X, Y, Z are positive, pair-wise relatively prime integers, where X is even, and Y, Z are odd [1-8].Open Science Journal – December 2017Open Science Journal Research ArticleEquation (I.1) shows that the squares of these variables,, must coincide with the uniquely defined solutions of Pythagoras’ equation, which are determined by the following expressions (I.2)where s and u being the odd and the even positive parameters, respectively [4], subject to inequality s > u and (s, u)=1. It is convenient to write Fermat’s equation with exponent n= 4 in the form (I.1) The variables X, Y, Z are positive, pair-wise relatively prime integers, where X is even, and Y, Z are odd [1-8]. Must coincide with the uniquely defined solutions of Pythagoras’ equation, which are determined by the following expressions (I.2)
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