Triangle numbers are calculated taking each integer plus all the ones before it. The first triangle number is 1, the second is 1+2 or 3, and the third is 1+2+3 or 6.

If you take 8 times a triangle number plus 1, the result will be a perfect square. This number also will be the square of the triangle number's place doubled, plus one.

For example, 6 is third in the triangle number sequence. (1, 3, 6...) This means 8 times 6 plus 1 = 49 equals 3 times 2 plus 1, squared, or 7 squared.

Prove why this works.

Triangle number T(n) = n*(n+1)/2 (T(1)=1, T(2)=3 and so on).
Thus, 8*T(n)+1 = 4*n*(n+1)+1 = (2*n+1)^2 which is a perfect square.

Posted by SRay on 2005-03-22 16:13:38