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"On a Diophantine Equation" paper of Erdös, at some point it is said that it is well known that $C(n,2)=x^2$ has infinitely many integer solutions. I am just wondering the formula generating all possible $n \geq 2$,because I wonder whether they all come by integer solutions of $2x^2-y^2=\pm 1$ Pell's equation generated by fundamental unit, or is it possible to have different $n$'s that do not come from the solutions of the pell's equation.

For example $C(2,2)=1$ a square, $C(9,2)=36$ a square too, so the first two $n$'s are 2,9.

One can also see from the solutions of $2x^2-y^2=\pm 1$ pell's equation, by take $x=1, y=0$ for $n=2=2*1$ and take $x=2, y=3$ for $n=9=3^2$.

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What is $C(n, 2)?$ – Igor Rivin Dec 16 2011 at 15:32
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 I would guess it is $\dbinom{n}{2}$... – Pedro Martins Rodrigues Dec 16 2011 at 15:37
$n$ and $n-1$ are relatively prime. Voting to close. – fedja Dec 16 2011 at 15:41
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 @fedja, what is the relevance of $n$ and $n-1$ being relatively prime? The equation $C(n,2)=x^2$ rewrites to $(2n-1)^2 - 8x^2 = 1$, so all solutions come from solutions to Pell's equation. – Barry Cipra Dec 16 2011 at 16:02
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 I view it as $n(n-1)=2x^2$, so $n=y^2$ and $n-1=2z^2$ or vice versa, both options resulting in Pell's equations mentioned in the original post. Anyway, no matter how you think of it, it is pretty obvious. Hence the closing vote. – fedja Dec 16 2011 at 16:08
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closed as too localized by Felipe Voloch, Dror Speiser, fedja, Igor Rivin, Emil Jeřábek Dec 16 2011 at 16:42

1 Answer

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Another approach to the problem, at least for non-number theorists, is to ask Mathematica:

Select[Range[10000], IntegerQ[Sqrt[Binomial[#, 2]]] &]

and you find that the first examples are

{1, 2, 9, 50, 289, 1682, 9801}

Then go to the OEIS (https://oeis.org/) and input that. You find that this is sequence A055997 (https://oeis.org/A055997), and the OEIS response is together with generating functions, recurrence relations, citations, and more.

One of those references is to an article titled "Discovering the Square-Triangular Numbers", which seems a promising title. The citation is to the Fibonacci Quarterly, and a little googling finds that their old issues are online (http://www.fq.math.ca/list-of-issues.html), and this particular article (by Phil Lafer) is, too (http://www.fq.math.ca/Scanned/9-1/lafer.pdf).

The article reads nicely (thank you Phil Lafer), and the Pell equation $2x^2-y^2=1$ does indeed make a fundamental appearance.

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