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

Frequently asked questions

Does the double Wieferich condition already rule out most exponent pairs by itself?

It is extremely restrictive numerically (only six pairs are known up to enormous search bounds), but it does not by itself prove no solution exists in general — the six known double Wieferich pairs could in principle still need to be checked, and there is no proof that they are the only ones. Finishing the argument for every possible pair requires the further algebraic elimination in the next steps.

Source

Preda Mihăilescu, 2002

Primary cyclotomic units and a proof of Catalan's conjecture

Condensed summary

Step 4 of 7: Mihăilescu's 1999 leap: a Wieferich-type congruence with no class-number cost

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In plain words

Karl Inkeri had shown in the 1990s that any solution forces a strong congruence condition — but checking it in practice required computing class numbers of number fields, which is itself extremely expensive. Mihăilescu's first major contribution (1999) removed that obstacle: he showed that the exponents of any solution must satisfy $p^{q-1}\equiv1\pmod{q^2}$ , a self-contained numerical condition reminiscent of the rare 'Wieferich primes' from Fermat's Last Theorem, with no class number needed at all.

This single congruence (plus its mirror image swapping $p$ and $q$ ) is dramatically easier to check by computer, and it is the key that finally let researchers verify, computationally, that no small exponent pair works.

p^{q-1}\equiv 1\pmod{q^2}\qquad(\text{and symmetrically}\ q^{p-1}\equiv1\pmod{p^2})

Detailed analysis

Inkeri's criterion (1990s) related solubility of Catalan's equation to the class number of an auxiliary field $K_p$ related to $\mathbb{Q}(\zeta_p)$ — powerful in principle, but computing class numbers is very costly, limiting how far it could be pushed numerically (Bilu 2004, §3, Theorem 3.1). Mihăilescu's first paper (1999) proved a stronger, 'class-number-free' criterion: for any solution of Catalan's equation, $q^2\mid x$ and $p^{q-1}\equiv1\pmod{q^2}$ — the latter now called Wieferich's relation (Bilu 2004, §3, Theorem 3.2).

The proof (Bilu 2004, §3.1) works inside the cyclotomic field $K=\mathbb{Q}(\zeta_p)$ : using the Stickelberger ideal $I$ (a specific ideal of the group ring $\mathbb{Z}[G]$ , $G=\mathrm{Gal}(K/\mathbb{Q})$ , that is known by the classical Stickelberger theorem to annihilate the class group of $K$ ), Mihăilescu shows that for suitable $\theta\in(1-\sigma)I$ (with $\sigma$ complex conjugation), the algebraic number $(x-\zeta)^\theta$ is exactly a $q$ -th power up to a root of unity. Careful bookkeeping of this identity modulo $q^2$ then forces both $q^2\mid x$ and the congruence $p^{q-1}\equiv1\pmod{q^2}$ .

By the symmetry of Catalan's equation, the mirror statement $q^{p-1}\equiv1\pmod{p^2}$ also holds; a pair $(p,q)$ satisfying both is called a double Wieferich pair. This purely arithmetic, computationally cheap criterion is what let Mignotte and Roy verify by computer that any solution needs $\min\{p,q\}>10^7$ (Bilu 2004, §4.4) — but a solution's exponents must still, somehow, be entirely eliminated, which is the remaining task.

Terms in this step

Stickelberger's theorem: A classical theorem in cyclotomic number theory identifying an explicit ideal (the Stickelberger ideal) in the group ring of the Galois group of $\mathbb{Q}(\zeta_p)$ that is guaranteed to annihilate the ideal class group — turning certain non-principal ideals into principal ones after multiplication by elements of this ideal.
Double Wieferich pair: A pair of odd primes $(p,q)$ satisfying both $p^{q-1}\equiv1\pmod{q^2}$ and $q^{p-1}\equiv1\pmod{p^2}$ . Only six such pairs are known below very large search bounds, making the condition extremely restrictive.