Primes of the form m³ + k
Let $k$ be a fixed integer other than a (positive or negative) cube, and let $P(n)$ be the number of primes of the form $m^3 + k$ less than $n$. For a prime $q \equiv 1 \pmod 3$ not dividing $k$, let $(-k)_q = 1$ if $-k$ is a cubic residue of $q$ and $(-k)_q = -\tfrac12$ otherwise. Hardy and Littlewood found no earlier formulation of problems of this kind, the simplest being primes of the form $m^3 + 2$.
Problem. Prove that there are infinitely many primes of the form $m^3 + k$, and that $$ P(n) \sim \frac{n^{1/3}}{\log n} \prod_{q} \left(1 - \frac{2}{q-1} (-k)_q\right) \qquad (n \to \infty), $$ the product running over the primes $q \equiv 1 \pmod 3$ that do not divide $k$.
Source: G. H. Hardy and J. E. Littlewood, Partitio Numerorum III, Acta Math. 44 (1923), Conjecture K.
Imported from doi.org/10.1007/BF02403921.
Classification
- Primary
- NT — Number Theory
- Keywords
- cubic residues, 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}
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00928, "Primes of the form m³ + k", clopen.solutions.