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.

The stipulation is equivalent to:

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

(2d+1)^2 = 8a+8b+1 [1]
b=d/2(d+1)-a, so

(2c+1)^2 = 4a^2+4a+4b^2+4b+1
(2c+1)^2 = 4a^2+4a+4(-a+d^2/2+d/2)^2+4(-a+d^2/2+d/2)+1
(2c+1)^2 = 8a^2-4ad(d+1)+(d^2+d+1)^2

Put back b:
4ad(d+1)-8a^2 = 8ab

when:
8ab=(d^2+d+1)^2-(2c+1)^2 [2]

and there are as many solutions as one wishes, since appropriate  factors can always be found.

To get to Charlie's solutions, a=x, b=y, (x+y) = (d(d+1))/2, (T(x)+T(y)) = (c(c+1))/2

 

Edited on August 31, 2014, 6:50 am

Posted by broll on 2014-08-31 06:02:40