Worked solution: Mihăilescu's proof of Catalan's conjecture via cyclotomic fields (2002)

Step 6 of 7: The final algebraic weapon: Thaine's theorem on annihilators

In plain words

Everything so far has narrowed the possible exponent pairs $(p,q)$ to double Wieferich pairs, and eliminated one dangerous sub-case, but a genuine algebraic elimination of every remaining possibility was still missing until Mihăilescu's second, decisive paper (2002/2004). The final tool he needed was a 1988 theorem of Francisco Thaine about which elements of the group ring of a Galois group are guaranteed to 'annihilate' certain natural modules built from units and class groups of the real subfield of a cyclotomic field.

Mihăilescu shows that the double Wieferich relation from Step 4, fed into a module built from the relevant cyclotomic units, forces that module to be annihilated by more elements than Thaine's theorem allows unless the module is actually trivial — and a trivial module here means the hypothetical solution simply cannot exist.

\mathrm{ann}_{\mathbb{Z}[G]}(M) \supseteq I(1-\sigma) \implies M = 0

Detailed analysis

Thaine's theorem (1988) identifies an explicit annihilator: elements built from the same Stickelberger-type combinatorics as before are shown to annihilate not just the class group (as Stickelberger's classical theorem says) but a related, more refined module built from the group of cyclotomic units modulo actual units in the real subfield $\mathbb{Q}(\zeta_p+\zeta_p^{-1})$ (Bilu 2004, §5, citing Thaine's 'On the ideal class groups of real abelian number fields'). This is strictly stronger than Stickelberger's theorem and was the missing ingredient no earlier attack on Catalan's equation had exploited.

Mihăilescu's central technical achievement (Bilu 2004, §6, reducing Theorem 1.3 to three further technical statements proved in the paper's final sections) is to construct, from a hypothetical solution $(x,y,p,q)$ , a specific $\mathbb{Z}[G]$ -module $M$ built from cyclotomic units and to show — using the double Wieferich relation of Step 4 together with $p\not\equiv1\pmod q$ from Step 5 — that the annihilator of $M$ must contain more of the group ring than Thaine's theorem permits for a nonzero module of that particular shape. The only way to avoid a contradiction is if $M$ is the zero module, which forces $x=\pm1$ or an equivalent degeneracy — impossible for a genuine solution with all of $x,y,p,q$ nontrivial.

This is, in outline, why no double Wieferich pair $(p,q)$ (and hence no exponent pair at all) can support a solution of $x^p-y^q=1$ : the arithmetic of cyclotomic units, controlled via Stickelberger's and Thaine's theorems, is simply too rigid to allow it. Mihăilescu announced this result in 2002 and published the full argument in Crelle's journal in 2004.

Terms in this step

Thaine's theorem: A 1988 theorem of Francisco Thaine strengthening Stickelberger's theorem: it identifies group-ring elements that annihilate not just the class group but a finer module measuring the gap between cyclotomic units and all units in the maximal real subfield of $\mathbb{Q}(\zeta_p)$ .
Cyclotomic units: A specific, explicitly describable subgroup of the units of a cyclotomic field (or its real subfield), built from expressions like $(1-\zeta^a)/(1-\zeta)$ ; comparing this subgroup to the full unit group is a classical way of measuring the field's class number.