OP-00926 Open

Every large non-square is a prime plus a square

Let $N(n)$ be the number of representations $n = p + m^2$ with $p$ prime and $m \ge 0$ an integer, and let $\bigl(\frac{n}{q}\bigr)$ be the Legendre symbol. Hardy and Littlewood found nothing in the literature on this question, probably because small exceptions such as $34$ and $58$ refute the claim for every number, even with $1$ counted as a prime.

Problem. Prove that every large number $n$ is either a square or the sum of a prime and a square, and that $$ N(n) \sim \frac{\sqrt n}{\log n} \prod_{q \ge 3} \left(1 - \frac{1}{q-1} \left(\frac{n}{q}\right)\right), $$ the product running over the odd primes.

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

Imported from doi.org/10.1007/BF02403921.

Classification

Primary: NT — Number Theory
Keywords: additive prime number theory, Hardy–Littlewood conjectures, sums of a prime and a square
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{davenport1937,
  author = {Davenport, H. and Heilbronn, H.},
  title = {Note on a result in the additive theory of numbers},
  journal = {Proceedings of the London Mathematical Society (2)},
  volume = {43},
  pages = {142--151},
  year = {1937}
}

@article{brunner1989,
  author = {Br{\"u}nner, R. and Perelli, A. and Pintz, J.},
  title = {The exceptional set for the sum of a prime and a square},
  journal = {Acta Mathematica Hungarica},
  volume = {53},
  pages = {347--365},
  year = {1989}
}

@article{perelli1995,
  author = {Perelli, A. and Zaccagnini, A.},
  title = {On the sum of a prime and a $k$-th power},
  journal = {Izvestiya: Mathematics},
  volume = {59},
  number = {1},
  pages = {189--204},
  year = {1995}
}

@misc{zhao2026,
  author = {Zhao, Genheng},
  title = {The exceptional set of a prime and a square},
  year = {2026},
  eprint = {2609.24051},
  archivePrefix = {arXiv},
  note = {Preprint; claims $E(X) = O(X^{4/5})$ for the exceptional set}
}

Citing this problem

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OP-00926, "Every large non-square is a prime plus a square", clopen.solutions.