%0 Journal Article
%T New Asymptotic Results on Fermat-Wiles Theorem
%A Kimou Kouadio Prosper
%A Kouakou Kouassi Vincent
%A Tano&#233
%A Fran&#231
%A ois
%J Advances in Pure Mathematics
%P 421-441
%@ 2160-0384
%D 2024
%I Scientific Research Publishing
%R 10.4236/apm.2024.146024
%X We analyse the Diophantine equation of Fermat &lt;i&gt;x&lt;/i&gt;&lt;i&gt;&lt;sup&gt;p&lt;/sup&gt;&lt;/i&gt;   &lt;i&gt;y&lt;/i&gt;&lt;i&gt;&lt;sup&gt;p&lt;/sup&gt;&lt;/i&gt; = &lt;i&gt;z&lt;/i&gt;&lt;i&gt;&lt;sup&gt;p&lt;/sup&gt;&lt;/i&gt; with &lt;i&gt;p&lt;/i&gt; &gt; 2 a prime, &lt;i&gt;x&lt;/i&gt;, &lt;i&gt;y&lt;/i&gt;, &lt;i&gt;z&lt;/i&gt; positive nonzero integers. We consider the hypothetical solution (&lt;i&gt;a&lt;/i&gt;, &lt;i&gt;b&lt;/i&gt;, &lt;i&gt;c&lt;/i&gt;) of previous equation. We use Fermat main divisors, Diophantine remainders of (&lt;i&gt;a&lt;/i&gt;, &lt;i&gt;b&lt;/i&gt;, &lt;i&gt;c&lt;/i&gt;), 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 &lt;i&gt;z&lt;/i&gt; &amp;#8211; &lt;i&gt;y&lt;/i&gt; = 1 then there exists a tight bound &lt;i&gt;N&lt;/i&gt; such that, for all prime exponents &lt;i&gt;p&lt;/i&gt; &gt; &lt;i&gt;N&lt;/i&gt; , we have &lt;i&gt;x&lt;/i&gt;&lt;i&gt;&lt;sup&gt;p&lt;/sup&gt;&lt;/i&gt;   &lt;i&gt;y&lt;/i&gt;&lt;i&gt;&lt;sup&gt;p&lt;/sup&gt;&lt;/i&gt; &amp;#8800; &lt;i&gt;z&lt;/i&gt;&lt;i&gt;&lt;sup&gt;p&lt;/sup&gt;&lt;/i&gt;.
%K Fermat&amp
%K #8217
%K s Last Theorem
%K Fermat-Wiles Theorem
%K Kimou&amp
%K #8217
%K s Divisors
%K Diophantine Quotient
%K Diophantine Remainders
%K Balzano Weierstrass Analysis Theorem
%U http://www.scirp.org/journal/PaperInformation.aspx?PaperID=133682