The squares of triangular numbers 1 and 6 are triangular numbers 1 and 36.
T1^2 = 1 * 1 = 1 = T1
T3^2 = 6 * 6 = 36 = T8
Are there additional triangular numbers whose squares form a triangular number?
(In reply to re(2): Solution
I don't think you are quite out of the woods yet. There are two holes in the proof.
Problem 1: It is not, for instance, correct that 14^2*15^2 can only be split into a product of two squares in one way. It equals 7^2*30^2 and 42^2*5^2 and 1^2*210^2. Also, note that 3^2*4^2 = 6^2*2^2, where both of the squares are even. However, one of the terms must be roughly twice the other, so this can probably be dealt with.
Problem 2: You are assuming that either 2b and (b+1) are squares, or b and 2(b+1) are squares, but this is also not proved. It does turn out to be true, however. Because b and (b+1) are relatively prime, any prime factor greater than 2 is necessarily represented entirely in one term or another. But that it is not necessarily the case with the factor of 2. 2b and (b+1), or b and 2(b+1), could conceivably both be 2 times a square. But if 2b and (b+1) are both 2 times a square, then b and 2(b+1) are both squares. And if b and 2(b+1) are both 2 times a square, then 2b and (b+1) are both squares. So it is always true that either 2b and (b+1) are squares, or b and 2(b+1) are squares.
So now I am agreeing with your conclusion, Harry. I have fixed problem 2, and a little hand-waving can be used to fix problem 1.
Also, I feel something more elegant just out of reach. Something about the problem 2 argument seems like it might get us directly to 'a' only being 1 or 3.