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).

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)