Catalan's conjecture

Remarks

The result is also called Mihăilescu's theorem: Eugène Catalan conjectured it in 1844, and Preda Mihăilescu proved it in 2002. Robert Tijdeman had shown in 1976, with Baker's theorem on linear forms in logarithms, that the Diophantine equation has only finitely many solutions. Mihăilescu's proof reduces to odd prime exponents and then uses the arithmetic of cyclotomic fields. Equivalently: $8$ and $9$ are the only two consecutive integers that are both perfect powers. The hypothesis $a, b > 1$ is needed, since for $b = 1$ every $x^a - 1$ is a valid $y$. Euler had settled the special case $x^2 - y^3 = 1$, whose only positive solution is $(3, 2)$.

Going deeper

For a student

The equation $x^a - y^b = 1$ is an exponential Diophantine equation, with unknown exponents. Tijdeman's finiteness result is typical of Baker's method: it bounds the solutions, but the bound was far too large to search. Mihăilescu's proof avoids computation and works with units and class groups of cyclotomic fields, a theme shared with work on Fermat's Last Theorem. The more general Pillai conjecture, that each positive integer occurs only finitely often as a difference of perfect powers, remains open.