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

,

i.e.,

.

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

.)

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