OP-00924 Open

Primes represented by a quadratic polynomial am² + bm + c

Let $a, b, c$ be integers with $a > 0$, $(a, b, c) = 1$, $a + b$ and $c$ not both even, and $D = b^2 - 4ac$ not a square. Let $P(n)$ be the number of primes less than $n$ of the form $am^2 + bm + c$ with $m \ge 0$ an integer, let $\varepsilon = 1$ if $a + b$ is odd and $\varepsilon = 2$ if $a + b$ is even, and let $$ C = \prod_{q \ge 3,\ q \nmid a} \left(1 - \frac{1}{q-1} \left(\frac{D}{q}\right)\right), $$ the product running over the odd primes not dividing $a$. Hardy and Littlewood explain each excluded case: if $(a, b, c) = d > 1$ there is at most one such prime; if $a + b$ and $c$ are both even, every value is even; and if $D$ is a square, the product $C$ diverges to zero.

Problem. Prove that there are infinitely many primes of the form $am^2 + bm + c$, and that $$ P(n) \sim \frac{\varepsilon C}{\sqrt a} \frac{\sqrt n}{\log n} \prod_{p} \frac{p}{p-1} \qquad (n \to \infty), $$ the product running over the odd primes $p$ dividing both $a$ and $b$.

Source: G. H. Hardy and J. E. Littlewood, Partitio Numerorum III, Acta Math. 44 (1923), Conjecture F.

Imported from doi.org/10.1007/BF02403921.

Classification

Primary: NT — Number Theory
Keywords: Bunyakovsky conjecture, Hardy–Littlewood conjectures, primes represented by polynomials
Literature checked: Oct. 25, 2026 by clemens-admin

References

@article{hardy1923,
  author = {Hardy, G. H. and Littlewood, J. E.},
  title = {Some problems of `{P}artitio numerorum'; {III}: {O}n the expression of a number as a sum of primes},
  journal = {Acta Mathematica},
  volume = {44},
  pages = {1--70},
  year = {1923},
  doi = {10.1007/BF02403921}
}

@article{bateman1962,
  author = {Bateman, Paul T. and Horn, Roger A.},
  title = {A heuristic asymptotic formula concerning the distribution of prime numbers},
  journal = {Mathematics of Computation},
  volume = {16},
  pages = {363--367},
  year = {1962}
}

@article{iwaniec1978,
  author = {Iwaniec, Henryk},
  title = {Almost-primes represented by quadratic polynomials},
  journal = {Inventiones Mathematicae},
  volume = {47},
  pages = {171--188},
  year = {1978}
}

@article{baier2007,
  author = {Baier, Stephan and Zhao, Liangyi},
  title = {Primes in quadratic progressions on average},
  journal = {Mathematische Annalen},
  volume = {338},
  pages = {963--982},
  year = {2007}
}

@article{baier2009,
  author = {Baier, Stephan and Zhao, Liangyi},
  title = {On primes in quadratic progressions},
  journal = {International Journal of Number Theory},
  volume = {5},
  number = {6},
  pages = {1017--1035},
  year = {2009}
}

Citing this problem

This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.

OP-00924, "Primes represented by a quadratic polynomial am² + bm + c", clopen.solutions.