TITLE:
Diophantine Quotients and Remainders with Applications to Fermat and Pythagorean Equations
AUTHORS:
Prosper Kouadio Kimou, François Emmanuel Tanoé
KEYWORDS:
Diophantine Equation, Modular Arithmetic, Fermat-Wiles Theorem, Pythagorean Triplets, Division Theorem, Division Algorithm, Python Program, Diophantine Quotients, Diophantine Remainders
JOURNAL NAME:
American Journal of Computational Mathematics,
Vol.13 No.1,
March
31,
2023
ABSTRACT:
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.
(1)
when , it is well known that this equation has an infinity of
solutions but has none (non-trivial) when . 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 non-trivial triplet () solution
of Equation (1) such
that . and are called the
Diophantine quotients and remainders of solution . We
compute the remainder and the quotient of b and c by a using the division algorithm. Hence, we have: and et with . We prove the following important results. if and only
if and if and only if . Also, we deduce that or for any hypothetical
solution . 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
properties to directly prove a partial result of FLT.