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

$A_s=a_2-a_4+a_6-\mathrm{\cdots }+a_{s-2}-a_s$,

$B_s=b_2-b_4+b_6-\mathrm{\cdots }+b_{s-2}-b_s$.

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