Consider the quadruplets (p,q,r,s) of positive integers with p>q>r>s, and satisfying pr+qs= (q+s+p-r)(q+s-p+r).
Is it ever the case that pq+rs is a prime number?
After looking over the raw data, I noticed that for the values I looked at that gcd(p+s, q-r) always divided pq+rs. A brute force search confirms this through 700>=p. I still dont know how to prove it though.