%0 Journal Article
%T On Fermat Last Theorem: The New Efficient Expression of a Hypothetical Solution as a Function of Its Fermat Divisors
%A Prosper Kouadio Kimou
%J American Journal of Computational Mathematics
%P 82-90
%@ 2161-1211
%D 2023
%I Scientific Research Publishing
%R 10.4236/ajcm.2023.131002
%X <p style=\"text-align:justify;\">
	Denote by <img src=\"Edit_c5b72c11-10d8-46ae-8f3f-92311ab006f9.png\" alt=\"\" /><span \"=\"\"> a non-trivial
primitive solution of Fermat¡¯s equation</span><span \"=\"\"> (<i>p</i> prime).We introduce, for the first
time, what we call Fermat principal divisors <img src=\"Edit_cb4e6625-1bc9-497c-8831-f002aa8e6162.png\" alt=\"\" /> of the triple </span><span \"=\"\"><img src=\"Edit_0e30b827-a49e-4af4-9607-3d15abb127aa.png\" alt=\"\" /></span><span \"=\"\"> defined as follows. <img src=\"Edit_1d759100-3f0b-4bbe-92ea-bd789e226d62.png\" alt=\"\" />, <img src=\"Edit_dd90d186-9495-42b3-b86d-c4756d23129e.png\" alt=\"\" /> and <img src=\"Edit_25c68698-378f-4920-898f-c96097211ea7.png\" alt=\"\" />. We
show that it is possible to express </span><span \"=\"\"><i>a</i>,<i>b</i> and <i>c</i> as function of the Fermat principal divisors. Denote by <img src=\"Edit_7e451d33-c7cc-4715-8cdb-3a9ed83cd17e.png\" alt=\"\" />the set of possible non-trivial solutions of the Diophantine equation <img src=\"Edit_c69ab09b-82d9-4530-80b4-e4917525c260.png\" alt=\"\" />.
And, let</span><span \"=\"\"><i></i><i></i><i></i><sub></sub><sub></sub> </span><span style=\"line-height:97%;font-family:;\" \"=\"\"><img src=\"Edit_da2e842a-ca77-47a1-8f93-baaf0963c8cb.png\" alt=\"\" /></span><span style=\"line-height:97%;font-family:;\" \"=\"\"></span><span \"=\"\"> </span><span \"=\"\">(<i>p</i> prime). We prove that, in the first
case of Fermat¡¯s theorem, one has</span> 
</p>
<p style=\"text-align:center;\">
	<span \"=\"\"><img src=\"Edit_3c742a9a-c0f4-4153-bd88-017a5e4d8d1e.png\" alt=\"\" /></span><span \"=\"\"> </span><span \"=\"\">.</span> 
</p>
<span><span><span><span style=\"font-family:;\" \"=\"\">In
the second case of Fermat¡¯s theorem, we show that <span></span></span></span></span></span> 
<p style=\"text-align:center;\">
	<br />
</p>
<p style=\"margin-left:10.0pt;\">
	<br />
</p>
<div style=\"text-align:center;\">
	<img src=\"Edit_04a66f36-a25f-4f08-b3de-01649e6e8b19.png\" alt=\"\" />, <img src=\"Edit_0fbb22ea-53d0-4fad-b164-97bfa42beaf8.png\" width=\"184\" height=\"124\" alt=\"\" />,<img src=\"Edit_de59928e-627c-4382-ad5d-c03611b13e18.png\" alt=\"\" />.
</div>
<span><span><span style=\"font-family:;\" \"=\"\">Furthermore,
we have implemented a python program to calculate the Fermat divisors of
Pythagoreans triples. The results of this program, confirm the model used. We
now have an effective tool to directly process Diophantine equations and that
of Fermat.</span></span></span> 
<p>
	<br />
</p>
%K Fermat¡¯s Last Theorem
%K Fermat Divisors
%K Barlow¡¯s Relations
%K Greatest Common Divisor
%U http://www.scirp.org/journal/PaperInformation.aspx?PaperID=123068