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After watching John Baez talking about the number 24 I got interested in the question: For which $n \in \mathbb{N}$ is there a number $m \in \mathbb{N}$ s.t.

$$ \sum_{k=1}^nk^2=m^2 $$

Or to put it in plain english, when is the sum of the first $n$ square numbers again a square.

The surprising answer to this is that this is only the case for $n=0,1,24$ and you proof this (according to him) with the formula

$$ \sum_{k=1}^nk^2=\dfrac{n(n+1)(2n+1)}{6} $$

and the machinery of elliptic curves. My guess is that you probably want to proof that the elliptic curve $E$ given by the above equation satisfies $E(\mathbb{Q})=\mathbb{Z}_3$ (not sure here, but something along these lines).

My main question is: can somebody name a reference for a proof of this? But I would also be interested if you could shed some light on how to proceed here or how much machinery you actually need to prove this.

Edit: Since the question got closed while I was not on a computer I just wanted to point out that the other thread doesn't answer my question. The answers in the other thread answers the question: For which $n$ are there $n$ consecutive squares that sum to a square. Note that there the consecutive squares don't have to start with 1. Part of the answer is that this is always the case when $n$ is a perfect square coprime to 6 which clearly is not the case in my situation. The problem is that the answer doesn't shed light on the number $a$ in the question. In particular it is not clear when $a=0$ and when not (which would be my situation). I'm not sure why so many people mark this as duplicate when the questions are clearly different.

Maik Pickl
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  • Thanks for pointing this out, I will have a look at it. – Maik Pickl Oct 07 '16 at 14:49
  • You are welcome. Actually, Thomas Andrews proof, which is very detailed, shows that you can also use more elementary techniques. – Dietrich Burde Oct 07 '16 at 14:57
  • @DietrichBurde I don't have time to read the answer in detail right now but isn't the question a little different? He is answering the question: When are there $n$ consecutive squares that sum to a square? But he is not necessarily starting at 1 which would leave my question open. Correct me if I'm wrong. – Maik Pickl Oct 07 '16 at 15:01
  • It also works for starting at $1$. By the way, here is an answer using elliptic curves. – Dietrich Burde Oct 07 '16 at 15:03
  • @DietrichBurde I don't see how Thomas answer in the other thread works for me, but maybe I'm just short-sigthed right now. But the link you provided looks interesting. Thanks a lot! – Maik Pickl Oct 07 '16 at 15:05
  • As an introductory reference on elliptic curves and their integer points, I like Tate and Silverman's Rational Points on Elliptic Curves. Fairly self-contained and written at an advanced undergraduate level. – hardmath Oct 07 '16 at 15:05
  • @hardmath Thanks for the reference. Looks very promising. I would still be interested in a reference where this particular example is discussed. – Maik Pickl Oct 08 '16 at 12:00
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    Yes, it seems to me the identified "duplicate" doesn't provide an answer to your Question. I'm voting to reopen. – hardmath Oct 08 '16 at 12:10
  • @DietrichBurde Have you actually read both answers? The one in the thread and the one you linked? I don't think you "can use this methods" to prove the case $a=0$ since these methods only provide the existence of solutions to the equation $x^2-ny^2=\frac{n(n-1)(n+1)}{3}$ and say nothing about $a=\frac{y-n+1}{2}-1$. And the other link also doesn't provide a full answer, what you easily see if you only look at the end of the answer provided. Sorry if I sound a little annoyed but to me it looks like you just glanced over the answer. I get that you don't want to have duplicated questions to... – Maik Pickl Oct 08 '16 at 15:56
  • avoid reduntant work and I appreciate your effort but from my perspective the answer is still open. Anyway, as it stands I will just figure it out myself. Thanks! – Maik Pickl Oct 08 '16 at 15:58
  • I am sorry. You are right, the duplicate is not explicitly solving this, on the other hand you can find so many references which solve this problem. If you want, here is a link: "It was shown by Watson (and again by Ljunggren) that if $0^2 + 1^2 + ... + n^2$ is a square then $n = 0, 1$ or $24$. The OEIS site has all links available. Click on Watson, for example. See also here. – Dietrich Burde Oct 08 '16 at 18:14
  • @DietrichBurde That is indeed a better duplicate! There is even a reference provided. Thanks again, this answers the question. :) – Maik Pickl Oct 08 '16 at 18:55

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