Determine analytically all pairs of positive rational numbers (p, q) satisfying p-q = p/q such that p+q is a positive integer.

If p and q are rational, and p + q is an integer, then they must be of the form p = a/d, q = b/d

Substituting and solving for a yields

a = b*b/(b-d)

b-d must divide b evenly if a is an integer, so let

b-d = k, and b = ek, for any integral k> 0, e > 1

then d = b - k = (e-1)*k

Subsituting into p = a/d, q = b/d

Gives :

p= e*e/(e-1) and q = e/(e-1) for all integral e > 0

But p + q must be integral

P + q = e*(e+1)/(e-1)

which is only an integer for integral e if e = 2 or 3

So, I get (4,2) and (9/2,3/2) as the only two solutions