Catalan's conjecture

Catalan's conjecture, now proven and known as Catalan's theorem, states that the only two perfect powers (numbers obtained by raising a whole number to an exponent greater than 1) that differ by exactly 1 are 8 and 9. In other words, the only solution to the equation \( x^a - y^b = 1 \) with integers \( x, y > 1 \) and exponents \( a, b > 1 \), is \( 3^2 = 9 \) and \( 2^3 = 8 \). This means no other pairs of perfect powers differ by just 1. The theorem was proven by mathematician Preda Mihăilescu in 2002.