Define T(N) as the Nth triangular number.

Each of X and Y is a positive integer such that:
Each of T(X)+T(Y) and X+Y is a triangular number.

Does there exist an infinite number of pairs (X,Y) that satisfy the given conditions? Give reasons for your answer.

(In reply to re: As many as you want. by Harry)

I agree; both equations [1] and [2] are needed.

The point I was trying to make is that equation [2] is pretty much at large, with 2 independent variables on either side. Given that degree of freedom in [2], it must follow that there exist an unlimited number of solutions to [2] that also satisfy [1].

I felt that this was point was clearer from the equations I offered rather than the starting point of:

a(a+1)+b(b+1) = c(c+1)
(a+b) = d/2(d+1)

For this reason I posted my answer as 'thoughts' rather than 'solution' or 'possible solution'.

Edited on September 2, 2014, 1:50 am

Posted by broll on 2014-09-02 01:49:43