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
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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) |
Bunyakovskii conjecture. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bunyakovskii_conjecture&oldid=46173