# 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 $N>x,y$,

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

Title Fermat’s last theorem (analytic form of) FermatsLastTheoremanalyticFormOf 2013-03-22 16:17:34 2013-03-22 16:17:34 whm22 (2009) whm22 (2009) 8 whm22 (2009) Theorem msc 11D41