Math Conjectures

Bateman–Horn Conjecture

OPENLandmarkConjectureProposed 1962 · Standard version

Canonical statement

Let f1,…,fk∈Z[x]f_1,\ldots,f_k\in\mathbb Z[x] be distinct irreducible polynomials with positive leading coefficients. Assume their product f=∏ifif=\prod_i f_i has no fixed prime divisor, meaning that no prime divides f(n)f(n) for every n∈Zn\in\mathbb Z. For a prime pp, set Np=#{a∈Z/pZ:f(a)=0}N_p=\#\{a\in\mathbb Z/p\mathbb Z:f(a)=0\}, and
C(f1,…,fk)=∏p1−Np/p(1−1/p)k. C(f_1,\ldots,f_k)= \prod_p\frac{1-N_p/p}{(1-1/p)^k}.
Then, as x→∞x\to\infty,
#{n∈Z:1≤n≤x, f1(n),…,fk(n) are all prime}∼C(f1,…,fk)∏i=1kdeg⁡fi∫2xdt(log⁡t)k. \#\{n\in\mathbb Z:1\le n\le x,\ f_1(n),\ldots,f_k(n)\text{ are all prime}\} \sim \frac{C(f_1,\ldots,f_k)} {\prod_{i=1}^k\deg f_i} \int_2^x\frac{dt}{(\log t)^k}.
View source LaTeX
Let \(f_1,\ldots,f_k\in\mathbb Z[x]\) be distinct irreducible polynomials with positive leading coefficients. Assume their product \(f=\prod_i f_i\) has no fixed prime divisor, meaning that no prime divides \(f(n)\) for every \(n\in\mathbb Z\). For a prime \(p\), set \(N_p=\#\{a\in\mathbb Z/p\mathbb Z:f(a)=0\}\), and
\[
  C(f_1,\ldots,f_k)=
    \prod_p\frac{1-N_p/p}{(1-1/p)^k}.
\] Then, as \(x\to\infty\),
\[
 \#\{n\in\mathbb Z:1\le n\le x,\
        f_1(n),\ldots,f_k(n)\text{ are all prime}\}
 \sim
 \frac{C(f_1,\ldots,f_k)}
      {\prod_{i=1}^k\deg f_i}
   \int_2^x\frac{dt}{(\log t)^k}.
\]
NT-PRIME-009:[BatemanHorn1962][Granville2008BH][FriedlanderIwaniec1998]
Number theoryAnalysis and functional analysis

Notes

The Bateman–Horn conjecture predicts, for distinct irreducible integer polynomials f1,…,fkf_1,\ldots,f_k with positive leading coefficients whose product has no fixed prime divisor, how often the values f1(n),…,fk(n)f_1(n),\ldots,f_k(n) are simultaneously prime: the count up to xx should be asymptotic to an explicit singular-series constant, normalized by the degrees, times ∫2x(log⁡t)−k dt\int_2^x(\log t)^{-k}\,dt. Bateman and Horn formulated this heuristic asymptotic in 1962 [BatemanHorn1962].

The conjecture unifies much of prime-counting lore: it contains Bunyakovsky's conjecture (a single polynomial), the prime kk-tuple predictions including twin primes, and, qualitatively, Schinzel's hypothesis HH, recorded in this catalog as a strong relative rather than a duplicate. The heuristic sits in the Cramér tradition of probabilistic models for the primes [Granville2008BH]. The settled cases involve a single linear polynomial, where Dirichlet's theorem and the prime number theorem for progressions apply; for a one-variable polynomial of degree at least 22 nothing is proved in this generality. The theorem of Friedlander and Iwaniec that X2+Y4X^2+Y^4 captures its primes shows what current methods reach in a thinner, two-variable setting [FriedlanderIwaniec1998].

The full conjecture, including its predicted constant, remains open.

References (3)

The boxed statement is the canonical open formulation — not a stronger variant or a related research program. The status reflects the catalog's last review; do your own literature search before investing serious effort.