Fermat’s last theorem (analytic form of)

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:

Theorem If $a_{n}=b_{n}$ for some odd integer $n>2$ , then either

(i) $A_{N}>0$ for some $N>x,y$ ,

(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

Title	Fermat’s last theorem (analytic form of)
Canonical name	FermatsLastTheoremanalyticFormOf
Date of creation	2013-03-22 16:17:34
Last modified on	2013-03-22 16:17:34
Owner	whm22 (2009)
Last modified by	whm22 (2009)
Numerical id	8
Author	whm22 (2009)
Entry type	Theorem
Classification	msc 11D41