Catalan's Conjecture

Abstract

It may be observed that 8 and 9 are consecutive numbers in this sequence. The first problem is: Are there any other consecutive integers in the above sequence? How many pairs of consecutive integers? Finitely many? Infinitely many? I may also consider the sequence of all proper powers, which includes also 5th powers, 7th powers, llth powers, etc... (note that powers with even exponents are squares, powers with exponent multiples of 3 are cubes . . . ) The same question may be asked: Are there consecutive powers other than 8 and 9? But for the sequence of all powers, a new problem makes sense: Are there three consecutive integers that are proper powers? Since powers grow very fast, lists of powers are necessarily very limited and, besides 8 and 9, no consecutive powers have ever been observed. This is an indication to keep in mind, but one should be careful before jumping to any conclusion. Just think, for example, that up to 100, 10% of the numbers are squares, up to 10,000, 1% are squares, up to 1,000,000, 1 in 1,000 are squares, and so on. Yet, Lagrange proved that despite the increasing scarcity of squares, every natural number is the sum of at most 4 squares. The squares seem to occupy strategic positions. Of course, ours is a different problem. Similar problems may be asked with the following sequence. Let a, b be integers, 1 < a < b and consider the sequence of all powers of a or of b. For example, if a = 2, b = 3, it is the sequence