View source for Bunyakovskii conjecture

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111040/b1110401.png" /> be a [[Polynomial|polynomial]] of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111040/b1110402.png" /> with integer coefficients. Already in 1854, V. Bunyakovskii [[#References|[a1]]] considered the problem whether <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111040/b1110403.png" /> represents infinitely many prime numbers as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111040/b1110404.png" /> ranges over the positive integers (cf. [[Prime number|Prime number]]). There are some obvious necessary conditions, e.g., that the coefficients of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111040/b1110405.png" /> are relatively prime, that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111040/b1110406.png" /> is irreducible (cf. [[Irreducible polynomial|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111040/b1110407.png" /> one has

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111040/b1110408.png" /></td> </tr></table>

Replacing the constant term <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111040/b1110409.png" /> by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111040/b11104010.png" /> with a suitable integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111040/b11104011.png" />, one can make <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111040/b11104012.png" /> irreducible, say with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111040/b11104013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111040/b11104014.png" />, etc. Hence, one has to assume that the values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111040/b11104015.png" /> for positive integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111040/b11104016.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111040/b11104017.png" /> represents infinitely many prime numbers. Similarly, the [[Dirichlet theorem|Dirichlet theorem]] about infinitely many primes in an arithmetic progression comes from considering the polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111040/b11104018.png" /> with relatively prime integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111040/b11104019.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111040/b11104020.png" />.

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

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

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  V. Bouniakowsky [V. Bunyakovskii],   "Sur les diviseurs numériques invariables des fonctions rationelles entières"  ''Mém. Sci. Math. et Phys.'' , '''VI'''  (1854–1855)  pp. 307–329</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  A. Schinzel,   W. Sierpiński,   "Sur certaines hypothèses concernant les nombres premiers"  ''Acta Arithm.'' , '''4'''  (1958)  pp. 185–208</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  H. Halberstam,   H.-E. Richert,   "Sieve methods" , Acad. Press  (1974)</TD></TR></table>