Article Dans Une Revue American Journal of Mathematics Année : 2023

Diophantine Quotients and Remainders with Applications to Fermat and Pythagorean Equations

Quotients et Diviseurs Diophantiens appliqués à l'Equation de Pythagore-Fermat

Résumé

Diophantine equations have always fascinated mathematicians about existence, finitude, and the calculation of possible solutions. Among these equations, one of them will be the object of our research. This is the Pythagoras’-Fermat’s equation defined as follows. x^n+y^n=z^n, n>=2 an integer (1) when n = 2 , it is well known that this equation has an infinity of solutions but has none (non-trivial) when n > 2 . We also know that the last result, named Fermat-Wiles theorem (or FLT) was obtained at great expense and its understanding remains out of reach even for a good fringe of professional mathematicians. The aim of this research is to set up new simple but effective tools in the treatment of Diophantine equations and that of Pythagoras-Fermat. The tools put forward in this research are the properties of the quotients and the Diophantine remainders which we define as follows. Let (a,b,c) a non-trivial triplet ( abc ≠ 0 ) solution of Equation (1) such that ar1 and q2=q1+1if and only if 2 1 r1>r2. Also, we deduce that q2=q1 or q2=q1+1 for any hypothetical solution (a,b,c). We illustrate these results by effectively computing the Diophantine quotients and remainders in the case of Pythagorean triplets using a Python program. In the end, we apply the previous proprties to directly prove a partial result of FLT.
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hal-04099438 , version 1 (23-06-2023)

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  • HAL Id : hal-04099438 , version 1

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Kouadio Prosper Kouadio Kimou, François Emmanuel Tanoé. Diophantine Quotients and Remainders with Applications to Fermat and Pythagorean Equations. American Journal of Mathematics, 2023, 13 (199-210), pp.199 - 210. ⟨hal-04099438⟩

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