Arbitrarily large arithmetic progressions only with perfect powers?

Question

How can I show that there are arbitrarily large arithmetic progressions consisting of perfect powers?

I think I could use the Chinese Remainder Thm here, but how can I use it? How can I organize the system of congruences?

Answer 1

It is possible to get arbitrarily long arithmetic progressions of perfect powers. The proof is by induction. Suppose you have an arithmetic progression of perfect powers of length $k$:

$$x_{1}^{n_{1}},...,x_{k}^{n_{k}}.$$

Let the common difference be $d$. Let $N=\operatorname{lcm}(n_{1},...n_{k})$. Then

$$(x_{k}^{n_{k}}+d)^{N}x_{1}^{n_{1}},\dots,(x_{k}^{n_{k}}+d)^{N}x_{k}^{n_{k}},(x_{k}^{n_{k}}+d)^{N+1}$$

is an arithmetic progression of perfect powers of length $k+1$.

For the basis case just consider any two perfect powers.

Answer 2

This is not a complete answer, but is too long for a comment, and I suspect that a complete answer to your question might be unattainable within current mathematical knowledge.

Firstly, a couple of known negative results:

1. It is not possible to have four (distinct) perfect squares in arithmetic progression. This was believed to be true by Fermat and proved by Euler.
2. It is not possible to have three (distinct) perfect $k$-th powers in arithmetic progression i.e. the three terms have the same exponent). This was proved by Darmon and Merel in 1997.

There are a number of other results of a similar nature, mostly of a negative nature, and you can get some good ideas of work done in this field by looking at some of the papers by Lajos Hajdu which are available here, as well as other papers referred to in his papers.

In particular, the paper "Arithmetic Progressions of Squares, Cubes and n-th Powers", would seem to make it very unlikely that you could find arbitrarily long arithmetic progressions, especially by any methods as simple as the Chinese Remainder Theorem.

Finally, I would be surprised in the Chinese Remainder Theorem were to be of much use in finding the kind of arithmetic sequences that you are looking for since, since (a) the CRT will help you find multiples of powers, but not numbers which are precisely exact powers, and (b) if the CRT could be used for general powers, I suspect it could be used to find arithmetic sequences of squares and cubes, which would contradict the negative results quoted above.