[science_man_88]

Originally Posted by carpetpool

Help, comments, suggestions appreciated.

For three integers p, q, r such that gcd(p, q) = 1, gcd(r, q) = 1, let d = pr.

(p+q)*(q+d) - pq = x

(q+d)*q - p*(p+q) = y

Prove that

r*x-q-d = y

so x=p(q)+r(p^2)+q^2+q(p)(r) -p(q) = d(p)+q^2+q(d)
and y=q^2+q(d)-p^2-p(q)
therefore y+q+d = q^2+q(d)-p^2+p(q)+q+d = (q+1+p)(q)+(q+1)(d)-p^2

r(x) = (q+1+p)(q)+(q+1)(d)-p^2 = d^2+r(q^2)+r(q)(d)

anyways for now I'm bored I guess.