New Asymptotic Results on Fermat-Wiles Theorem

Abstract

We analyse the Diophantine equation of Fermat <i>x</i><i>^p</i> <i>y</i><i>^p</i> = <i>z</i><i>^p</i> with <i>p</i> > 2 a prime, <i>x</i>, <i>y</i>, <i>z</i> positive nonzero integers. We consider the hypothetical solution (<i>a</i>, <i>b</i>, <i>c</i>) of previous equation. We use Fermat main divisors, Diophantine remainders of (<i>a</i>, <i>b</i>, <i>c</i>), an asymptotic approach based on Balzano Weierstrass Analysis Theorem as tools. We construct convergent infinite sequences and establish asymptotic results including the following surprising one. If <i>z</i> &#8211; <i>y</i> = 1 then there exists a tight bound <i>N</i> such that, for all prime exponents <i>p</i> > <i>N</i> , we have <i>x</i><i>^p</i> <i>y</i><i>^p</i> &#8800; <i>z</i><i>^p</i>.