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

Step 3 of 7: Tijdeman's 1976 breakthrough: solutions are effectively bounded

In plain words

For over a century after Catalan's letter, nobody could even show the equation had only finitely many solutions, let alone find them all. In 1976 Robert Tijdeman changed this completely, using Alan Baker's deep theory of linear forms in logarithms of algebraic numbers to prove that the exponents $p,q$ — and hence, via Cassels' relations, the bases $x,y$ too — must be smaller than some specific, explicitly computable (if gigantic) number.

This turned Catalan's conjecture from a genuinely open question into, in principle, a finite calculation — though one far too large to carry out directly. It set the stage for a decades-long race to shrink Tijdeman's bound down to something checkable.

\text{Baker's theorem} \implies |x|,|y|,p,q \le \text{effective constant}

Detailed analysis

Baker's theory (from the 1960s) gives explicit lower bounds for $|\Lambda|$ where $\Lambda=b_1\log\alpha_1+\cdots+b_n\log\alpha_n$ is a nonzero 'logarithmic form' in algebraic numbers $\alpha_i$ with integer coefficients $b_i$ — bounds strong enough to beat the trivial estimate one gets just from $\Lambda\ne0$ (Bilu 2004, §4.1). Tijdeman's insight (1976) was to apply this to logarithmic forms built from Catalan's equation itself, such as $\Lambda=q\log|y+1|-p\log|x|$ , using Cassels' relations to control the auxiliary quantities involved (Bilu 2004, §4.2–4.3).

Comparing the resulting lower bound against a trivial upper bound derived directly from the equation forces an inequality between $p$ and $q$ (roughly $p\ll q\log q\log p$ and, symmetrically, $q\ll (\log p)^2\log q$ ), which together give an absolute, effective upper bound on both exponents — and hence, via Cassels again, on $x,y$ as well (Bilu 2004, §4.2, eqs. 15–17). Explicit numerical work by Langevin (1977), and later O'Neil and Mignotte, brought this bound down over the years to concrete numbers like $p\le 7.8\times10^{16}$ .

This is a genuine milestone — Catalan's problem is now 'decidable' in principle — but Tijdeman's bound alone is far too large to check by computer. The breakthrough that actually finishes the proof needs a sharper, purely algebraic idea, which is Mihăilescu's contribution in the next step.

Terms in this step

Linear forms in logarithms (Baker's theory): A branch of transcendence theory, pioneered by Alan Baker in the 1960s, that gives explicit lower bounds on how close a nonzero combination $b_1\log\alpha_1+\cdots+b_n\log\alpha_n$ (integer $b_i$ , algebraic $\alpha_i$ ) can come to zero — turning many Diophantine equations into problems with effectively computable solution bounds.