Catalan's conjecture

The only solution in integers $x, y, a, b$ with $a, b > 1$ of the equation $x^a - y^b = 1$ is $x=3, a=2, y=2, b=3$ — that is, $3^2 - 2^3 = 1$, so 8 and 9 are the only two consecutive perfect powers of positive integers.

Eugène Charles Catalan posed the conjecture in an 1844 letter to Crelle's Journal, after noticing 8 and 9 seemed to be the only consecutive perfect powers among small numbers. Progress was slow: in 1976 Robert Tijdeman used Baker's theory of linear forms in logarithms to show that, if any solution besides $3^2-2^3=1$ exists, both exponents $a$ and $b$ must be bounded by an explicit (astronomically large) constant, reducing the problem to a finite — but computationally hopeless — search. Preda Mihăilescu found the final, purely algebraic argument in 2002, using properties of cyclotomic fields and Wieferich pairs to prove no other solution exists, without relying on Tijdeman's huge numerical bound; the result is now called Mihăilescu's theorem.