FermatsLastTheoremAnalyticFormOf 2006-10-02 15:10:57 2006-10-12 12:21:14 Theorem % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here Let $x$, $y$, $z$ be positive real numbers. For each positive integer $r$, let $a_r = (x^r+y^r)/r!$ and $b_r=z^r/r!$. For $s$ divisible by 4, let $A_s=a_2-a_4+a_6- \cdots +a_{s-2}-a_s$, $B_s=b_2-b_4+b_6- \cdots +b_{s-2}-b_s$. Then Fermat's last theorem is equivalent (by elementary means) to: {\bf Theorem} If $a_n=b_n$ for some odd integer $n>2$, then either (i) $A_N > 0$ for some $N>x,y$, \hspace{2mm} or (ii) $B_M>0$ for some $M>z$. For a proof that these theorems are equivalent see: proof of equivalence of Fermat's Last Theorem to its analytic form