Problem
NT-B2-M12-P013 A Polynomial Cannot Always Give Primes
#13
★★★★☆ Level 4 of 5
Let \(f\in\mathbb Z[x]\) be a nonconstant polynomial. Prove that it is impossible for all numbers \(f(1),f(2),f(3),\ldots\) to be positive primes.
Take one prime value \(p=f(a)\) and consider \(f(a+pt)\).
Let \(p=f(a)\) be a positive prime. For every integer \(t\), \(a+pt\equiv a\pmod p\), hence \(f(a+pt)\equiv f(a)\equiv0\pmod p\). Since \(f\) is nonconstant, \(|f(a+pt)|\) is unbounded as \(t\) grows. For sufficiently large \(t\), we get a positive value greater than \(p\) but divisible by \(p\). It is composite, a contradiction.
Olympiad move: fix one prime value and generate new values divisible by it.