TITLE: About the infinite solutions of a Diophantine equation
QUESTION [7 upvotes]: Consider the following problem:
$$\sum_{k=1}^N k^2=q^2$$
where q is an integer number. This can be written as:
$$\frac{1}{3}N^3+\frac{1}{2}N^2+\frac{1}{6}N=q^2$$
In the same way we can write:
$$\sum_{k=1}^N k^3=q^3$$
that means:
$$\frac{1}{4}N^4+\frac{1}{2}N^3+\frac{1}{4}N^2=q^3$$
In general, we can consider the following equation:
$$\sum_{k=1}^N k^r=q^r$$
with $r$ integer number.
The question is: does this equation has infinite solutions for every $r$ integer?
Thanks.

REPLY [3 votes]: Generally, Schaffer proved in 1956 that the equation
$$
\sum_{k=1}^K k^r = q^n
$$
has at most finitely many solutions unless 
$$
(r,n) \in \{ (1,2), (3,2), (3,4), (5,2) \}.
$$
The proof uses Siegel's theorem (though, as Erick notes, Faltings would lead to the same conclusion).