Bunyakovsky conjecture

The Bunyakovsky conjecture (or Bouniakowsky conjecture) gives a criterion for a polynomial $f(x)$ in one variable with integer coefficients to give infinitely many prime values in the sequence $f(1),f(2),f(3),\ldots.$ It was stated in 1857 by the Russian mathematician Viktor Bunyakovsky. The following three conditions are necessary for $f(x)$ to have the desired prime-producing property: The leading coefficient is positive, The polynomial is irreducible over the rationals (and integers), and The values $f(1),f(2),f(3),\ldots$ have no common factor larger than 1. (In particular, the coefficients of $f(x)$ should be relatively prime.) Bunyakovsky's conjecture is that these conditions are sufficient: if $f(x)$ satisfies (1)–(3), then $f(n)$ is prime for infinitely many positive integers $n$.