I've just started reading Ribenboim's "13 lectures on FLT" and in it he makes sure to first explain the statement of the theorem and point out that, effectively, it has to be proven only for the exponents $4$ and $p$, where $p$ is an odd prime. This was clear to me, but after this he goes on to point out 2 further "cases" of the theorem:
The statement of Fermat's last theorem is often subdivided further into
two cases:The first case holds for the exponent $p$ when there do not exist integers
$x,y,z$ such that $p\nmid xyz$ and $x^p+y^p=z^p$. The second case holds for the exponent $p$ when there do not exist integers $x,y,z$, all different from $0$, such that $p\mid xyz$, $gcd(x,y,z) = 1$ and $x^p+y^p=z^p$.
If I understood these statements correctly, the goal is to make the distinction for the case when $p$ divides none of the three terms (the first case) and the case when it divides only one (the second case). Why is it necessary to do this? What effect on the problem does the amount of terms divisible by the exponent have?
Best Answer
"It is necessary to do this" only in the approach "à la Kummer". Recall that Kummer's work (starting from around 1845) amounts roughly to the theory of ideals in the ring $\mathbf Z[\zeta _p]$, where $\zeta _p$ is a primitive $p$-th root of unity. In your notations, Kummer showed that if $p$ is regular, i.e. $p$ does not divide the class number of $\mathbf Q(\zeta _p)$, then FLT holds true. Looking at the details of the proof (see e.g. Washington's book "Introduction to cyclotomic fields", chap.9), you'll see that it requires two distinct properties : 1) $p$ does not divide the class number of $\mathbf Q(\zeta _p)^+$ (the maximal totally real subfield of $\mathbf Q(\zeta _p)$) ; 2) under a certain congruential condition, a unit is a $p$-th power in $\mathbf Q(\zeta _p)^+$. The second property is a theorem (op. cit. thm.5.36), whose proof is immediate in the so called first case, but not in the second.
It is not necessary to do this in the "modular" approach, where FLT is just a corollary of the Shimura-Taniyama-Weil conjecture proved by Wiles around 1995: Every elliptic curve defined over $\mathbf Q$ is modular (in very vague terms, "comes from" a modular form). The distinction between Kummer's two cases becomes irrelevant. The elliptic curve coming into play is the Hellegouarch-Frei curve : starting from an hypothetical non trivial solution of Fermat's equation $a^p+b^p=c^p$, consider the elliptic curve $E$ defined by $y^2=x(x+\epsilon a^p)(x-\epsilon b^p),\epsilon=\pm 1$. Around 1969, Y. Hellegouarch showed that it has so many good properties that it could not possibly exist (he called it "Orlando's mare", from Ariosto's epic poem "Orlando Furioso"), but he could not go further because the theory of modular forms was not developped enough at that time. Much later, around 1985, G. Frey had the brilliant intuition that "Orlando's mare" could not be a modular curve, and this was proved by K. Ribet. The rest is history.