Find all positive integers n which make the expression below an integer:

(25/2 + (625/4 - n)^½ )^½ + (25/2 - (625/4 - n)^½ )^½

Well, the square of the expression above is 25 +2n^½

Set k = the square root of 25 +2n^½

Then n = ((k*k - 25)/2)^2

This is positive and integral whenever k is an odd number > 5

k = 7 --> n = 12*12 = 144
k = 9 --> n = 28*28 = 784
k = 11 --> n = 48*48 = 2304

etc.

Posted by Steve Herman on 2009-03-20 13:14:55