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Catalan's conjecture
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Catalan's conjecture

Catalan's conjecture is a simple conjecture in number theory that was proposed by the mathematician Eugène Charles Catalan.

To understand the conjecture notice that 23 = 8 and 32 = 9 are two consecutive powerss of natural numbers. Catalan's conjecture states that this is the only case of two consecutive powers.

That is to say, Catalan's conjecture states that the only solution in the natural numbers of

xa − yb = 1
for x,a,y,b > 1 is x = 3, a = 2, y = 2, b = 3.

In particular, notice that it's unimportant that the same numbers 2 and 3 are repeated in the equation 32 − 23 = 1. Even a case where the numbers were not repeated would still be a counterexample to Catalan's conjecture.

Catalan's conjecture was proved by Preda Mihăilescu; in April 2002, so it is now a theorem. The proof was checked by Yuri Bilu and makes extensive use of the theory of cyclotomic fields and Galois modules.

Pillai's conjecture concerns a general difference of perfect powers. It states that the differences in the sequence of all perfect powers tend to infinity, so that each given difference occurs only finitely many times. It is an open problem as of 2004 and is named for S. S. Pillai.

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