# Almost Composite

If p and q are primes, and x^{2} – px + q = 0 has distinct positive integer roots, find p and q.

- 5 August, 2012 -
- Article, Math Problems -
- Tags : 104 problems, number theory, titu
- 0 Comments

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.