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?

If pq+rs is prime, then the factors on the right side must be 1 and pq+rs.

Since q+s+p-r > q+s+r-p, we obtain the system of equations

q+s+p-r = pq+rs  and  q+s+r-p = 1. 

Couple that with the third condition that GCD(pq, rs)=1 and we can probably figure this out.

Posted by Tommy on 2007-06-28 14:23:15