Bunyakovskii conjecture

Let be a polynomial of degree with integer coefficients. Already in 1854, V. Bunyakovskii [a1] considered the problem whether represents infinitely many prime numbers as ranges over the positive integers (cf. Prime number). There are some obvious necessary conditions, e.g., that the coefficients of are relatively prime, that 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 one has

Replacing the constant term by with a suitable integer , one can make irreducible, say with , , etc. Hence, one has to assume that the values for positive integers 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 represents infinitely many prime numbers. Similarly, the Dirichlet theorem about infinitely many primes in an arithmetic progression comes from considering the polynomial with relatively prime integers and .

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).

References

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