Asymptotic count of primes that are sums of three positive cubes
Note: The first assertion, infinitely many such primes, follows from Heath-Brown's theorem that infinitely many primes have the form x³ + 2y³ with x, y positive (Acta Math. 186, 2001), as he notes; the asymptotic formula is open.
Let $P(n)$ be the number of primes of the form $k^3 + l^3 + m^3$ less than $n$, with $k, l, m$ positive integers, each prime counted as often as it is so represented. For a prime $q \equiv 1 \pmod 3$, let $\rho = e^{2\pi i/3}$, let $\omega = u + v\rho$ be the prime factor of $q$ in $\mathbb Z[\rho]$ with $u \equiv -1$ and $v \equiv 0 \pmod 3$, and let $A = 2u - v$, so that $4q = A^2 + 27B^2$ with $3B = v$.
Problem. Prove that there are infinitely many primes of the form $k^3 + l^3 + m^3$ with $k, l, m$ positive, and that $$ P(n) \sim \Gamma(4/3)^3 \frac{n}{\log n} \prod_{q} \left(1 - \frac{A}{q^2}\right) \qquad (n \to \infty), $$ the product running over the primes $q \equiv 1 \pmod 3$.
Source: G. H. Hardy and J. E. Littlewood, Partitio Numerorum III, Acta Math. 44 (1923), Conjecture N.
Imported from doi.org/10.1007/BF02403921.
Classification
- Primary
- NT — Number Theory
- Keywords
- cubic residues, Hardy–Littlewood conjectures, primes represented by polynomials, sums of cubes
- 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{heathbrown2001,
author = {Heath-Brown, D. R.},
title = {Primes represented by $x^3+2y^3$},
journal = {Acta Mathematica},
volume = {186},
pages = {1--84},
year = {2001}
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00931, "Asymptotic count of primes that are sums of three positive cubes", clopen.solutions.