Let’s say we want to find all integers such that
Suppose that are sufficiently large. Having spotted the solution , re-write the equation as
So . Checking the powers of 3 modulo 16 one deduces that . Writing therefore gives
Hence . As before, one deduces that . (A better way: since , we have by this.) Writing gives
Using , it follows that must be even. Then divides the left-hand side, which is impossible since the left-hand side is always odd. Thus we conclude that is the only solution.
This kind of approach usually works for many similar equations. The basic idea is the following:
- Guess an initial solution.
- Use the initial solution to group terms and factor them.
- Use congruence relations to restrict the variables.
- Eventually one side will likely have a prime factor that does not divide the other side.
This motivates the following:
Conjecture. Given integers with , the equation
has only finitely many solutions in integers.
This turns out to be a special case of a more general conjecture known as Pillai’s conjecture.
(All this arose when I was trying to make as small as possible with respect to , which range through the positive integers satisfying