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

Frequently asked questions

Why should a purely arithmetic equation like

x^p-y^q=1

have anything to do with cyclotomic fields?

Because $x^p-1$ factors completely over the cyclotomic field $\mathbb{Q}(\zeta_p)$ as $\prod_{i=0}^{p-1}(x-\zeta_p^i)$ . Writing Catalan's equation as $x^p-1=y^q$ and factoring the left side this way is exactly what lets Cassels — and later Mihăilescu — bring in the far more powerful machinery of algebraic number theory.

Source

Preda Mihăilescu, 2002

Primary cyclotomic units and a proof of Catalan's conjecture

Condensed summary

Step 2 of 7: Cassels' relations:

q

divides

x

, and

p

divides

y

Previous step Next step

In plain words

The first real leverage on Catalan's equation comes from a 1960 discovery of J.W.S. Cassels: if $x^p-y^q=1$ has a solution, the two exponents secretly cross-divide the two bases. Concretely, $q$ must divide $x$ and $p$ must divide $y$ — a strong and unexpected entanglement between the equation's four unknowns.

Cassels went further, showing precisely how $x-1$ and $y+1$ factor (into a $(q-1)$ -th or $(p-1)$ -th power of $p$ or $q$ , times an extra piece). These relations already force any hypothetical solution to be enormous, and they are the starting point for every later refinement.

x-1 = p^{q-1}a^q,\quad y = pav\qquad\text{and symmetrically}\qquad y+1=q^{p-1}b^p,\quad x=qub

Detailed analysis

Cassels (1960) proved that $q\mid x$ and $p\mid y$ , and moreover exhibited the precise factorizations $x-1=p^{q-1}a^q$ , $y=pav$ for some nonzero integer $a$ and positive integer $v$ , together with the symmetric relations $y+1=q^{p-1}b^p$ , $x=qub$ for some nonzero integer $b$ and positive integer $u$ (Bilu 2004, §2, Proposition 2.1). A short computation with $\zeta$ a primitive $p$ -th root of unity in the field $K=\mathbb{Q}(\zeta)$ then shows that the algebraic number $\eta=(x-\zeta)/(1-\zeta)$ is an algebraic integer whose principal ideal is a $q$ -th power of an ideal of $K$ (Bilu 2004, §2, Corollary 2.2) — the first hint that cyclotomic number theory, not just elementary arithmetic, is the right tool.

These relations already yield explicit, if weak, lower bounds: $|x|>p^{q-1}-1$ and $|y|>q^{p-1}-1$ follow immediately, and a refinement by Hyyrö (1964) sharpens this considerably (Bilu 2004, §2). Any hypothetical solution of Catalan's equation must therefore involve astronomically large numbers unless $p,q$ themselves are large — pointing towards the strategy of bounding, and eventually eliminating, the possible pairs $(p,q)$ .

The next steps sharpen this picture dramatically, first via effective bounds from transcendence theory (Tijdeman), then via a much finer arithmetic criterion discovered by Mihăilescu himself.

Terms in this step

Cyclotomic field $\mathbb{Q}(\zeta_p)$: The number field obtained by adjoining a primitive $p$ -th root of unity $\zeta_p$ to the rational numbers $\mathbb{Q}$ . Its arithmetic (units, ideals, class group) is far richer than that of $\mathbb{Q}$ itself, and is the setting for the entire rest of Mihăilescu's proof.