[Math] Show that the sum of the first $n$ natural numbers is a perfect square for infinitely many $n$.

Question

Show that the sum of the first $n$ natural numbers is a perfect square for infinitely many $n$.

The question doesn't make any sense to me. Any help is appreciated.

Answer 1

$1+2+\cdots + n = \dfrac{n(n+1)}{2}=k^2 \to n^2+n - 2k^2 = 0 \to \triangle = 1-4(1)(-2k^2) = 1+8k^2= m^2 \to m^2-8k^2 = 1$. This is a Pell equation with initial solution $(m,k) = (3,1)$, then it has infinitely many solutions, proving the claim.