Square-Triangle Numbers

We are asked to find all numbers $N = a ^2 = \frac{1}{2} ( b^2 + b )$ .

This can be manipulated into $(2b+1) ^2 - 2 (2a)^2 = 1$ .

This is a pell's equation , and since $(3, 2)$ is a starting solution, there are finitely many of them. The next few solutions are $(17,12), (577, 408), (19601, 13860) , \ldots$ .

Since we have $N = a^2$ , this corresponds to $1^2 , 6 ^2, 204^2, 6930^2 , \ldots$ , of which the first 2 terms match the 2 examples in the problem. These are not easy to find or guess directly, but easy once you know the underlying theory.

Perfect squares and triangle numbers are created from pre-existing numbers. Because this is the case, there are infinitely many triangle numbers, as well as perfect squares. Since there is an infinite amount of each it would be impossible if they overlapped finitely many times. Therefore, they must overlap infinitely many times. Therefore, there are infinitely many triangular numbers.

Note : I may be wrong here. I'm using probability as an ally. Imagine choosing a real number on a number line from 0-1. The probability of picking a rational number is essentially $\frac{1}{\infty}$ , or essentially 0. The same principle applies here (I think). Please correct me if I am wrong.