If p and q are primes, and x2 – px + q = 0 has distinct positive integer roots, find p and q.
Because is quadratic, it must have exactly two distinct roots; let’s call these roots .
Since are positive integer solutions to , then it must be the case that
Now, since is prime, then in order for , one of the two roots must be 1 or else becomes composite. Suppose , then is prime, and is also prime. Hence, are consecutive primes.
Due to the fact that all consecutive pairs must have exactly one even (even-odd or odd-even), then a consecutive pair of primes must contain a two. The only satisfying pair then is , giving us the quadratic
with solutions 1 and 2.