Bunyakovsky conjecture

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The Bunyakovsky conjecture (or Bouniakowsky conjecture) gives a criterion for a polynomial ${\displaystyle f(x)}$ in one variable with integer coefficients to give infinitely many prime values in the sequence${\displaystyle f(1),f(2),f(3),\ldots .}$ It was stated in 1857 by the Russian mathematician Viktor Bunyakovsky. The following three conditions are necessary for ${\displaystyle f(x)}$ to have the desired prime-producing property:

1. The leading coefficient is positive,
2. The polynomial is irreducible over the rationals (and integers), and
3. There is no common factor for all the infinitely many values ${\displaystyle f(1),f(2),f(3),\ldots }$. (In particular, the coefficients of ${\displaystyle f(x)}$ should be relatively prime. It is not necessary for the values f(n) to be pairwise relatively prime.)

Bunyakovsky's conjecture is that these conditions are sufficient: if ${\displaystyle f(x)}$ satisfies (1)–(3), then ${\displaystyle f(n)}$ is prime for infinitely many positive integers ${\displaystyle n}$.

A seemingly weaker yet equivalent statement to Bunyakovsky's conjecture is that for every integer polynomial ${\displaystyle f(x)}$ that satisfies (1)–(3), ${\displaystyle f(n)}$ is prime for at least one positive integer ${\displaystyle n}$: but then, since the translated polynomial ${\displaystyle f(x+n)}$ still satisfies (1)–(3), in view of the weaker statement ${\displaystyle f(m)}$ is prime for at least one positive integer ${\displaystyle m>n}$, so that ${\displaystyle f(n)}$ is indeed prime for infinitely many positive integers ${\displaystyle n}$. Bunyakovsky's conjecture is a special case of Schinzel's hypothesis H, one of the most famous open problems in number theory.