Catalan's problem

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*"[…the] following theorem, which I believe to be true, although I have not yet succeeded in proving it completely: two consecutive integers other than 8 and 9 cannot both be perfect powers; in other words, the equation x^m – yⁿ = 1, in which the unknowns are positive integers, has only one solution."*

A famous number-theory problem was born!

Progress toward proving this conjecture was slow and initially concerned special cases. As early as 1850, French mathematician Victor-Amédée Le Besgue (1791–1875) proved that, for every prime number p, the equation *x ^p – y*² = 1 has no nontrivial integer solutions.

More than a century then passed before Chinese mathematician Zhao Ke (1910–2002) proved the following theorem: if p is prime, then the trivial solutions 3² – 2³ = 1 and (–3)² – 2³ = 1 are the only solutions to x² – *y ^p* = 1.