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

Step 7 of 7: Closing the loop: Catalan's conjecture is a theorem

In plain words

Putting all the pieces together: any hypothetical counterexample to Catalan's conjecture would, after the reduction of Step 1, give odd primes $p,q$ and nonzero integers $x,y$ solving $x^p-y^q=1$ . Step 2's relations, Step 3's effective bound, Step 4's Wieferich congruences, Step 5's exclusion of $p\equiv1\pmod q$ , and Step 6's Thaine-theorem elimination together show this is impossible for every possible $(p,q)$ — with no exceptions and no remaining cases.

So the only solution to $x^u-y^v=1$ in integers greater than $1$ is exactly the one Catalan spotted in 1844: $3^2-2^3=1$ , i.e. $8$ and $9$ really are the only consecutive perfect powers. What began as an offhand remark in a letter to a journal editor took 158 years, and ultimately the deep arithmetic of cyclotomic fields, to settle.

3^2 - 2^3 = 1

Detailed analysis

Assembling Steps 1–6: suppose, for contradiction, that Catalan's equation $x^p-y^q=1$ has a solution in nonzero integers $x,y$ and odd primes $p,q$ . By Step 2, $q\mid x$ , $p\mid y$ , and $x,y$ satisfy Cassels' explicit factorizations. By Step 3, $p,q$ (and hence $x,y$ ) are bounded by an effective, if enormous, constant. By Step 4, $(p,q)$ must be a double Wieferich pair; by Step 5, additionally $p\not\equiv1\pmod q$ and $q\not\equiv1\pmod p$ . Step 6's module-theoretic argument, via Thaine's theorem, then shows that no pair $(p,q)$ satisfying all of the above can actually occur — the module $M$ built from the hypothetical solution is forced to be simultaneously nonzero (since $x,y,p,q$ are all nontrivial) and zero (by the annihilator computation), a contradiction (Bilu 2004, §6–9, culminating in the proof of Theorem 1.3).

Therefore Catalan's equation $x^p-y^q=1$ has no solution at all in nonzero integers and odd primes $p,q$ — this is Mihăilescu's Theorem 1.3, first announced in 2002 and published with full details in Journal für die reine und angewandte Mathematik (Crelle's journal) in 2004. Combined with Lebesgue's theorem ( $v=2$ case) and Ko Chao's theorem ( $u=2$ case) from Step 1, this establishes the original 1844 conjecture in full: $x^u-y^v=1$ has exactly one solution in integers $x,y,u,v>1$ , namely $3^2-2^3=1$ .

Catalan's conjecture — one of the longest-standing named conjectures in number theory to be resolved by a single mathematician's proof — is now properly called Mihăilescu's theorem.