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This question comes from trying to see why 24 is the only non-trivial value of $n$ for which $1^2+2^2+3^2+\cdots+n^2$ is a perfect square.
To this end, let $m,n \in \mathbb N$ be such that $1^2+2^2+3^2+\cdots+n^2 = m^2$, or $n(n+1)(2n+1) = 6m^2.$ When $n=24$ the left hand side is $24\times 25 \times 49$ and there are two things that make it work as a solution: $7^2+1=2\times5^2$ $7^2-1=12\times2^2.$ We can write these algebraically as $x^2=2y^2-1$ $x^2=12z^2+1$ and solve them simultaneously (with $x,y,z\in \mathbb N$).

These are instances of Pell's equation and each individually has an infinite number of solutions.
How do we show there is only one (non-trivial) value of $x$ that is common to the solutions of both equations?

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This problem is sometimes called cannonball problem or square pyramid problem.

Maybe you may have a look at MathWorld or Wikipedia and some references therein.

Elementary solution is given in W. S. Anglin: The Square Pyramid Puzzle, The American Mathematical Monthly, Vol. 97, No. 2 (Feb., 1990), pp. 120-124.

Further useful references:

Online resources:

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    @user236182 I have removed the paper from that website. (I should not have put it there in the first place.)2016-01-08