Posted by Jason Polak on 25. November 2017 · Write a comment · Categories: number-theory · Tags: ,

If you write out all the $n$th powers of natural numbers for $n \gt 1$ (“higher powers”) in increasing order you get a sequence like this:

1, 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 81, 100, 121, 125, 128, 144, 169, 196, 216, 225, …

If you keep computing more numbers in this sequence, once in a while you’ll find two consecutive terms quite close together. For instance:
$$1138^2 – 109^3 = 15$$
But no matter how far you go, it seems that the only two terms that are just one apart are 8 and 9. Eugène Catalan conjectured that 8 and 9 are the only two numbers in this sequence just one apart, and this conjecture became known as Catalan’s conjecture.

If you only allowed powers of 2 and 3 in the sequence, then the proof that 8 and 9 are the only consecutive powers is not difficult. In fact, it just requires the difference of squares identity: $a^2 – b^2 = (a – b)(a + b)$. Let’s see how it works! If only powers of 2 and 3 are allowed, then the question becomes: when is it possible that $3^m – 2^n = 1$ or that $2^n – 3^m = 1$?
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