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^24ad(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 20140831 06:02:40 