Every large odd number is a prime plus twice a square
Let $N(n)$ be the number of representations $n = p + 2m^2$ with $p$ prime and $m \ge 0$ an integer, and let $\bigl(\frac{2n}{q}\bigr)$ be the Legendre symbol. As Hardy and Littlewood note, Stern and his pupils (1856) verified that the only odd numbers below $9000$ not of this form are $5777$ and $5993$.
Problem. Prove that every large odd number $n$ is the sum of a prime and twice a square, and that $$ N(n) \sim \frac{\sqrt{2n}}{\log n} \prod_{q \ge 3} \left(1 - \frac{1}{q-1} \left(\frac{2n}{q}\right)\right), $$ the product running over the odd primes.
Source: G. H. Hardy and J. E. Littlewood, Partitio Numerorum III, Acta Math. 44 (1923), Conjecture I.
Imported from doi.org/10.1007/BF02403921.
Classification
- Primary
- NT — Number Theory
- Keywords
- additive prime number theory, Goldbach's other conjecture, Hardy–Littlewood conjectures
- 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}
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00927, "Every large odd number is a prime plus twice a square", clopen.solutions.