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Post Closed as "too localized" by Felipe Voloch, Dror Speiser, fedja, Igor Rivin, Emil Jeřábek

3 \$

"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\$.

2 cleaned up, tex, added tag

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

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

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

Thanks,$n=9=3^2$.

1

# Explicit solutions of C(n,2)=x^2 ?

"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>=2,because I wonder whether they all come by integer solutions of 2x^2-y^2=-+1 pell's equation generated by fundamental unit, or is it possible to have different n's that does 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=-+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,

Thanks,
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