Proofe that 1^2 + 2^2 + 3^2 ... + k^2 = n^2 has only one solution.

Factual Questions
1

I’ve been curious about this for a while, and was just reminded of it, so I thought I’d ask.

A square pyramidal number has the form 1[sup]2[/sup] + 2[sup]2[/sup] + 3[sup]2[/sup] + … + k[sup]2[/sup], which by induction is equivalent to k(k+1)(2k+1)/6. A square number has the form n[sup]2[/sup]. Apparently, it’s been proven that there is only one number (besides 1) which is both square and square-pyramidal, 4900=70[sup]2[/sup]=1[sup]2[/sup] + 2[sup]2[/sup] + 3[sup]2[/sup] + … + 24[sup]2[/sup].

Where can I find this proof? I’m currently in first year Calculus; is this proof something that I could understand?

This isn’t homework. I promise.

2

Just ignore the extra “e” in “proof” okay?

3

This page offers a couple of references to “elementary” proofs of this:

Anglin, W. S. “The Square Pyramid Puzzle.” Amer. Math. Monthly 97, 120-124, 1990.

Ma, D. G. “An Elementary Proof of the Solution to the Diophantine Equation 6y[sup]2[/sup]=x(x+1)(2x+1).” Sichuan Daxue Xuebao 4, 107-116, 1985.

I’m not familiar with either of these proofs, so I can’t say how understandable they would be to a first year calculus student.

(Note that “elementary” is not necessarily synonymous with “easy”. “Elementary” essentially means “from first principles”, and not involving higher mathematical concepts. The proof itself may still be fairly involved, logically).

If you Google “Cannonball Problem” you might even find a proof on the 'net, I’m not sure.

4

I managed to find one of them online at http://www.math.ubc.ca/~bennett/paper21.pdf (note: PDF) but my eyes started to glaze over by the second page. Still, it looks like something that I might be able to understand within a few months, so I’ll bookmark it and come back. Thank you!