Take the 2-minute tour ×  Kidding, kidding. But I do have a
question about an $n$-line outline of a proof of the first case
of FLT, with $n$ relatively small.  Here's a result of Eichler
(remark after Theorem 6.23 in Washington's Cyclotomic Fields):
If $p$ is prime and the $p$-rank of the class group of
$\mathbb{Q}(\zeta_p)$ satisfies $d_p<\sqrt{p}-2$, then the first
case of FLT has no non-trivial solutions. Once you know
Herbrand-Ribet and related stuff, the proof of this result is
even rather elementary.  The condition that $d_p<\sqrt{p}-2$
seems reminiscent of rank bounds used with Golod-Shafarevich to
prove class field towers infinite. More specifically, a possibly
slightly off (and definitely improvable) napkin calculation
gives me that for $d_p>2+2\sqrt{(p-1)/2}$, the $p$-th cyclotomic
field $\mathbb{Q}(\zeta_p)$ has an infinite $p$-class field
tower. It's probably worth emphasizing at this point that by the
recent calculation of Buhler and Harvey, the largest index of
irregularity for primes less than