布尼亚科夫斯基猜想1857

首项系数为正，

多项式在整数上是不可约的， f(1),f(2),f(3),… 的公因数为 1。

而布尼亚科夫斯基猜想这些条件就足够了：也就是说如果 f(x) 满足前面3点条件，则存在无限多个正整数 n 使 f(n) 是质数。

三个条件的讨论

第一个条件是必要的，因为如果首项系数是负的那么对所有够大的 x 都有 f(x)<0，特别的，对够大的正整数 n 都有 f(n) 是负数，从而非质数。（这里需要有素数为正的约定。）

第二个条件是必要的，因为如果 f(x)=g(x)× h(x)，其中 g(x)， h(x) 都是整系数多项式，那么由于 g(x) 和 h(x) 都只能有限次的等于-1，0，1，因此 f(x) 都有可能会是合数。

第三个条件是必要的，这也是显而易见的。

The Bunyakovsky conjecture (or Bouniakowsky conjecture) gives a criterion for a polynomial f(x) in one variable with integer coefficients to give infinitely many prime values in the sequence f(1),f(2),f(3),…. It was stated in 1857 by the Russian mathematician Viktor Bunyakovsky. The following three conditions are necessary for f(x) to have the desired prime-producing property:

The leading coefficient is positive,

The polynomial is irreducible over the rationals (and integers), and

The values f(1),f(2),f(3),… have no common factor larger than 1. (In particular, the coefficients of f(x) should be relatively prime.)

Bunyakovsky's conjecture is that these conditions are sufficient: if f(x) satisfies (1)–(3), then f(n) is prime for infinitely many positive integers n.

A seemingly weaker yet equivalent statement to Bunyakovsky's conjecture is that for every integer polynomial f(x) that satisfies (1)–(3), f(n) is prime for at least one positive integer n: but then, since the translated polynomial f(x+n) still satisfies (1)–(3), in view of the weaker statement f(m) is prime for at least one positive integer m>n, so that f(n) is indeed prime for infinitely many positive integers n. Bunyakovsky's conjecture is a special case of Schinzel's hypothesis H, one of the most famous open problems in number theory.

Discussion of three conditions

The first condition is necessary because if the leading coefficient is negative then f(x)<0 for all large x, and thus f(n) is not a (positive) prime number for large positive integers n. (This merely satisfies the sign convention that primes are positive.)

The second condition is necessary because if f(x)=g(x)h(x) where the polynomials g(x) and h(x) have integer coefficients, then we have f(n)=g(n)h(n) for all integers n; but g(x) and h(x) take the values 0 and ± 1 only finitely many times, so f(n) is composite for all large n.

The second condition also fails for the polynomials reducible over the rationals.

For example, the integer-valued polynomial P(x)=(1/12)· x⁴+(11/12)· x²+2 doesn't satisfy the condition (2) since P(x)=(1/12)· (x⁴+11x²+24)=(1/12)· (x²+3)· (x²+8), so at least one of the latter two factors must be a divisor of 12 in order to have P(x) prime, which holds only if |x|≤ 3. The corresponding values are 2,3,7,17, so these are the only such primes for integral x since all of these numbers are prime.

Examples

A simple quadratic polynomial

Some prime values of the polynomial f(x)=x²+1 are listed in the following table. (Values of x form OEIS sequence A005574; those of x²+1 form A002496.)

That n²+1 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

it is not known that this sequence extends indefinitely.

Cyclotomic polynomials

The cyclotomic polynomials Φ ₖ(x) for k=1,2,3,… satisfy the three conditions of Bunyakovsky's conjecture, so for all k, there should be infinitely many natural numbers n such that Φ ₖ(n) is prime. It can be shown that if for all k, there exists an integer n > 1 with Φ ₖ(n) prime, then for all k, there are infinitely many natural numbers n with Φ ₖ(n) prime.

The following sequence gives the smallest natural number n > 1 such that Φ ₖ(n) is prime, for k=1,2,3,…:

Partial results: only Dirichlet's theorem

To date, the only case of Bunyakovsky's conjecture that has been proved is that of polynomials of degree 1. This is Dirichlet's theorem, which states that when a and m are relatively prime integers there are infinitely many prime numbers （公式略）. This is Bunyakovsky's conjecture for f(x)=a+mx (or a-mx m<0).

The third condition in Bunyakovsky's conjecture for a linear polynomial mx+a is equivalent to a and m being relatively prime.

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.

Generalized Bunyakovsky conjecture

Given k≥ 1 polynomials with positive degrees and integer coefficients, each satisfying the three conditions, assume that for any prime p there is an n such that none of the values of the k polynomials at n are divisible by p. Given these assumptions, it is conjectured that there are infinitely many positive integers n such that all values of these k polynomials at x=n are prime. This conjecture is equivalent to the generalized Dickson conjecture and Schinzel's hypothesis H.