Catalan's conjecture

Catalan's conjecture (or Mihăilescu's theorem) is a theorem in number theory that was conjectured by the mathematician Eugène Charles Catalan in 1844 and proven in 2002 by Preda Mihăilescu at Paderborn University. The integers 23 and 32 are two perfect powers (that is, powers of exponent higher than one) of natural numbers whose values (8 and 9, respectively) are consecutive. The theorem states that this is the only case of two consecutive perfect powers. That is to say, that

Catalan's conjecture — the only solution in the natural numbers of

for a, b > 1, x, y > 0 is x = 3, a = 2, y = 2, b = 3.

For Catalan's aliquot sequence conjecture, see Aliquot sequence § Catalan–Dickson conjecture.
For Catalan's Mersenne number conjecture, see Double Mersenne number § Catalan–Mersenne number conjecture.
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