in one variable with integer coefficients to give infinitely many prime values in the sequence
It was stated in 1857 by the Russian mathematician Viktor Bunyakovsky.
to have the desired prime-producing property: Bunyakovsky's conjecture is that these conditions are sufficient: if
is prime for infinitely many positive integers
A seemingly weaker yet equivalent statement to Bunyakovsky's conjecture is that for every integer polynomial
is prime for at least one positive integer
still satisfies (1)–(3), in view of the weaker statement
is indeed prime for infinitely many positive integers
Bunyakovsky's conjecture is a special case of Schinzel's hypothesis H, one of the most famous open problems in number theory.
The first condition is necessary because if the leading coefficient is negative then
(This merely satisfies the sign convention that primes are positive.)
The second condition also fails for the polynomials reducible over the rationals.
This isn't a counterexample to Bunyakovsky conjecture since the condition (2) fails.
have gcd 1, is obviously necessary, but is somewhat subtle, and is best understood by a counterexample.
In practice, the easiest way to verify the third condition is to find one pair of positive integers
is written in the basis of binomial coefficient polynomials:
and the coefficients in the right side of the equation have gcd 2.
[citation needed] Some prime values of the polynomial
should be prime infinitely often is a problem first raised by Euler, and it is also the fifth Hardy–Littlewood conjecture and the fourth of Landau's problems.
Despite the extensive numerical evidence [2] it is not known that this sequence extends indefinitely.
satisfy the three conditions of Bunyakovsky's conjecture, so for all k, there should be infinitely many natural numbers n such that
It can be shown[citation needed] that if for all k, there exists an integer n > 1 with
prime, then for all k, there are infinitely many natural numbers n with
The following sequence gives the smallest natural number n > 1 such that
This case of Bunyakovsky's conjecture is widely believed, but again it is not known that the sequence extends indefinitely.
is prime,[citation needed] but there are exceptions; the first few are: To date, the only case of Bunyakovsky's conjecture that has been proved is that of polynomials of degree 1.
The third condition in Bunyakovsky's conjecture for a linear polynomial
No single case of Bunyakovsky's conjecture for degree greater than 1 is proved, although numerical evidence in higher degree is consistent with the conjecture.
polynomials with positive degrees and integer coefficients, each satisfying the three conditions, assume that for any prime
Given these assumptions, it is conjectured that there are infinitely many positive integers