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

We have already pointed out that pcbouhid's expression, when squared, equals 25 + 2n^½

Because the square of his expression is real and positive whenever n >= 0, we can conclude that the expression is real (not imaginary) whenever n >= 0, even if n is not an integer.  In other words, any imaginary parts of the two terms are guaranteed to cancel out when n >= 0.   We know this because the square of an imaginary number can never be real and positive.

Edited on March 27, 2009, 11:13 am

Posted by Steve Herman on 2009-03-27 00:07:21