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?

(In reply to

Helpful Observation? by Tommy)

At the outset, assume that (pq+rs) is prime.

Write, pq+rs = (p+s)r + (q-r)p = = t*gcd(p+s, q-r), where t is a positive integer as each of p, q, r and s are positive.

Thus, either t = 1, or gcd(p+s, q-r) = 1(why?)

Examine the two cases

*Edited on ***June 29, 2007, 12:57 pm**