# Bunyakovskii conjecture

Let $f ( x )$ be a polynomial of degree $\geq 1$ with integer coefficients. Already in 1854, V. Bunyakovskii [a1] considered the problem whether $f ( n )$ represents infinitely many prime numbers as $n$ ranges over the positive integers (cf. Prime number). There are some obvious necessary conditions, e.g., that the coefficients of $f$ are relatively prime, that $f$ is irreducible (cf. Irreducible polynomial) and, trivially, that the leading coefficient is positive. Are these conditions sufficient?

As Bunyakovskii remarked, the answer is "no" . For instance, for each prime number $p$ one has

$$n ^ {p} - n - p \equiv0 ( { \mathop{\rm mod} } p ) \textrm{ for all integers } n.$$

Replacing the constant term $p$ by $pk$ with a suitable integer $k$, one can make $x ^ {p} - x - pk$ irreducible, say with $p = 2$, $p = 3$, etc. Hence, one has to assume that the values $f ( n )$ for positive integers $n$ are not all divisible by a prime number. Bunyakovskii's conjecture is that these conditions are sufficient.

A special case of this conjecture is that the polynomial $x ^ {2} + 1$ represents infinitely many prime numbers. Similarly, the Dirichlet theorem about infinitely many primes in an arithmetic progression comes from considering the polynomial $ax + b$ with relatively prime integers $a > 0$ and $b$.

Bunyakovskii's conjecture was rediscovered and generalized to several polynomials by A. Schinzel [a2]; see also the comments in [a3].

P.T. Bateman and R. Horn have conjectured an asymptotic behaviour (cf. Bateman–Horn conjecture).

How to Cite This Entry:
Bunyakovskii conjecture. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bunyakovskii_conjecture&oldid=46173
This article was adapted from an original article by S. Lang (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article