OP-00923 Open

The Hardy–Littlewood asymptotic formula for primes of the form m² + 1

Let $P(n)$ be the number of primes of the form $m^2 + 1$ less than $n$, and let $$ C = \prod_{q \ge 3} \left(1 - \frac{1}{q-1} \left(\frac{-1}{q}\right)\right), $$ the product running over the odd primes, with $\bigl(\frac{-1}{q}\bigr)$ the Legendre symbol. Hardy and Littlewood report 301 such primes below $9\,000\,000$, against $302.6$ from the formula below (in an equivalent integral form).

Problem. Prove that there are infinitely many primes of the form $m^2 + 1$, and that $$ P(n) \sim C \frac{\sqrt n}{\log n} \qquad (n \to \infty). $$

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

Imported from doi.org/10.1007/BF02403921.

Classification

Primary: NT — Number Theory
Keywords: Hardy–Littlewood conjectures, Landau's problems, 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{iwaniec1978,
  author = {Iwaniec, Henryk},
  title = {Almost-primes represented by quadratic polynomials},
  journal = {Inventiones Mathematicae},
  volume = {47},
  pages = {171--188},
  year = {1978}
}

@article{deshouillers1982,
  author = {Deshouillers, J.-M. and Iwaniec, H.},
  title = {On the greatest prime factor of $n^2+1$},
  journal = {Annales de l'Institut Fourier},
  volume = {32},
  number = {4},
  pages = {1--11},
  year = {1982}
}

@article{merikoski2023,
  author = {Merikoski, Jori},
  title = {Largest prime factor of $n^2+1$},
  journal = {Journal of the European Mathematical Society},
  volume = {25},
  number = {4},
  pages = {1253--1284},
  year = {2023}
}

@article{friedlander1998,
  author = {Friedlander, John and Iwaniec, Henryk},
  title = {The polynomial $X^2+Y^4$ captures its primes},
  journal = {Annals of Mathematics},
  volume = {148},
  pages = {945--1040},
  year = {1998}
}

@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}
}

Citing this problem

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OP-00923, "The Hardy–Littlewood asymptotic formula for primes of the form m² + 1", clopen.solutions.