What sums of equal powers of consecutive natural numbers are powers of the same order? Trivially $n^1=n^1$, and everyone knows that $3^2+4^2=5^2$. Denis Serre quoted $3^3+4^3+5^3=6^3$ in a recent MathOverflow question (which prompted this one). Are any other examples known? 
 A: 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. 
A: Let S be the set of integers k such that there exists a sequence of k consecutive squares whose sum is a square.  According to the paper "Squares Expressible as Sum of Consecutive Squares" by L. Beekmans, S is known to be infinite and density 0; the citation is to problem 6552 in the American Math Monthly.
If F(x) is the sum of the first x squares, then you are really asking about integral solutions to the Diophantine equation
(*)  F(x) - F(y) = z^2
which is a double cover of the plane branched at a cubic curve (Even a reducible cubic curve, since x-y | F(x) - F(y).)  Heuristically, you would expect about N^{1/2} solutions as x and y range over a box of size N.  It would be interesting to ask:
a) whether the geometry of this surface is so easy to describe that you can say something about its integral points; and
b) whether (*) has a solution in polynomials in one variable (or, what's the same -- does the surface contain an affine line?)
A: Well, if we consider n consecutive 4th powers with initial a,
F(a,n) = a^4 + (a+1)^4 + (a+2)^4 + ... + (a+n-1)^4
or, equivalently,
F(a,n) = (n/30)(-1+30a^2-60a^3+30a^4+30a(1-3a+2a^2)n+10(1-6a+6a^2)n^2+(-15+30a)n^3+6n^4) 
it is easy to check that F(a,n) = y^4 (or even just y^2) has NO solution in the positive integers with BOTH {a,*n*} < 1000, with the exception of the trivial n = 1. (I had checked this with Mathematica some time back.)
If we relax your question and allow n 4th powers in arithmetic progression d equal to some kth power, then the smallest I found was 64 4th powers with common difference d = 2 starting with,
29^4 + 31^4 + 33^4 + ... + 155^4 = 96104^2
P.S. The closed-form formula for general d is available, but I find it too tedious to include in this post. 
A: There are 126 pairs $i\lt x\le 1000$ such that $i^2+(i+1)^2+...+x^2$ is a square. If you fix $i$ then the sum $i^2+...+x^2$ is a cubic polynomial $f_i(x)$ in $x$. So you are looking for integer points on the elliptic curve $y^2=f_i(x)$. For example for $i=3$,  the first of these are $(4,5), (580, 8075), (963,17267)$. I hope number theorists here can give more information. See also the comment by JSE below. 
A: J. C. Ottem's example $1^2 + ... + 24^2 = 70^2$ in the comments is of particular mathematical interest; it is one way to construct the Leech lattice, and is therefore somehow mysteriously related to other appearances of the number $24$ in mathematics (see, e.g. John Baez's thoughts).
A: While browsing the site http://sites.google.com/site/tpiezas/Home
mentioned in the comments above, I found this on the page for cubes:
"There are many particular cubic equations with this property, one of which is $9^3+13^3+19^3+23^3 = 28^3, (9+23 = 13+19) as well as those in a nice arithmetic progression like,
11^3+12^3+13^3+14^3 = 20^3
31^3+33^3+35^3+37^3+39^3+41^3 = 66^3" .
You might ask Mr. Piezas directly for more examples.
Gerhard "Ask Me About System Design" Paseman, 2011.01.24
