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

Solution by Harry)

Harry:

A very nice start, but there are holes in your solution. I am not yet convinced that your conclusion (that there are no others) is even true.

In your equation, a^{2}(a + 1)^{2} = 2b(b+1),

the LHS can be split into an odd number times an even number in more than one way.

For instance, 14^2*15^2

14 = 2*7 and 15 = 3*5, so there are many ways to express this product as an odd number times and even number.

Even worse, it could be expressed as an even number times an even number, which would set up several more equations.