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