Trivially $n^1=n^1$, and everyone knows that $3^2+4^2=5^2$. [Denis Serre](https://mathoverflow.net/questions/53048) quoted $3^3+4^3+5^3=6^3$ in a recent MathOverflow question (which prompted this one). Are any other examples known? 



What sums of equal powers of consecutive natural numbers are powers of the same order?
There is a good discussion at <http://www.mathpages.com/home/kmath147.htm> along with some nice examples, e.g., $6^3 + 7^3 + \dots + 69^3 = 180^3$, $1134^3 + \dots + 2133^3 = 16830^3$, which apparently are part of an infinite family (starting with $3^3+4^3+5^3=6^3$). There is a table of sums of consecutive cubes equal to a cube, not coming from this infinite family. The author of this page (Kevin Brown, if I'm not mistaken) writes, "If we go on to consider sums of higher powers, it appears that there are no sums of two or more consecutive 4th powers equal to a 4th power, or in general sums of two or more consecutive $n$th powers equal to an $n$th power for any $n\gt3$. Can anyone supply a proof, reference, or counter-example?" 


I suspect that any proof will be too big too fit in the margin. 


