2010/062) What are all the rational numbers x such that x + 1/x is an integer?

x + 1/x = (x^2 + 1)/x

This will be an integer if x^2 + 1 = kx, for some integer k. Equivalently,

x^2 - kx + 1 = 0

==> x = [ k +/- sqrt(k^2 - 4) ] / 2


for this to be an integer sqrt(k^2 - 4) has to be an integer say h( same parity as k that is both odd or both even)

k^2-4 = h^2

k^2 - h^2 = 4

so (k-h)(k+h) = 4 = 1 * 4 and 2* 2 and -1 * -4 and -2 * - 2

so k+ h = k-h = 2 or k-h = k+ h = - 2

k+ h = k-h = 2 => k = 2 giving x = 1

k+ h = k-h = 2 => k = -2 giving x = -1

so x = 1 or - 1