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I read in "Letters to a young mathematician" that 4900 is the only square integer that is the sum of consecutive squares (I believe he meant by that "starting from 1", but maybe that's not even necessary). I did a quick run through with a python script and of course this seems totally devoid of a computational pattern. Why is 4900 (and 1 of course) the only number such that this works?

I did find out that the sum of squares is the following...

$\sum^{n}_{i=1} i^2 = \frac{n(n+1)(2n+1)}{6}$

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    $\begingroup$ What is the question? $\endgroup$
    – Jonas Meyer
    Commented May 31, 2010 at 8:15
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    $\begingroup$ mathworld.wolfram.com/CannonballProblem.html $\endgroup$
    – KConrad
    Commented May 31, 2010 at 8:19
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    $\begingroup$ math.ubc.ca/~bennett/paper21.pdf $\endgroup$
    – Jonas Meyer
    Commented May 31, 2010 at 8:21
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    $\begingroup$ Jonas, you could just put that as the answer, that'll get me where I need to be. The CannonballProblem link doesn't help unless you have the papers. $\endgroup$
    – Michael Hoffman
    Commented May 31, 2010 at 8:36
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    $\begingroup$ It seems no one has picked up on the question of whether "starting from 1" is necessary. Of course, it is; trivially, every square integer is the sum of (one) consecutive square(s). Slightly less trivially, $5^2=3^2+4^2$, $29^2=20^2+21^2$ (the first two members of an infinite sequence of examples obtainable via Pell). There's also $38^2+39^2+\dots+48^2=143^2$, again the first in an infinite sequence of square sums of 11 consecutive squares, and many, many more. $\endgroup$
    – Gerry Myerson
    Commented Jun 30, 2010 at 2:55

2 Answers 2

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This is a classical Diophantine equation (Mordell, Diophantine Equations, p. 258). Apart from n = 0, 1, -1, there is only the solution n = 24. Proofs by G. N. Watson (1919), W. Ljunggren (1952).

There is some history in http://www.math.ubc.ca/~bennett/paper21.pdf .

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I remember vaguely that the equation $1^2+\dots+24^2=70^2$ gives a construction of the Leech lattice (one has to consider one of the two even neighbours of the lattice $\mathbb Z \frac{w}{70}+\Lambda'$ where $w=(1,2,\dots,24)$ and $\Lambda'=\lbrace z\in\mathbb Z^{24}|\langle z,w\rangle\in70\mathbb Z\rbrace$) which is a rather unique and exceptional structure.

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