Every large non-cube is a prime plus a cube
Let $N(n)$ be the number of representations $n = p + m^3$ with $p$ prime and $m$ a positive integer. For a prime $q \equiv 1 \pmod 3$ not dividing $n$, let $(n)_q = 1$ if $n$ is a cubic residue of $q$ and $(n)_q = -\tfrac12$ otherwise.
Problem. Prove that every large number $n$ is either a cube or the sum of a prime and a positive cube, and that $$ N(n) \sim \frac{n^{1/3}}{\log n} \prod_{q} \left(1 - \frac{2}{q-1} (n)_q\right), $$ the product running over the primes $q \equiv 1 \pmod 3$ that do not divide $n$.
Source: G. H. Hardy and J. E. Littlewood, Partitio Numerorum III, Acta Math. 44 (1923), Conjecture L.
Imported from doi.org/10.1007/BF02403921.
Classification
- Primary
- NT — Number Theory
- Keywords
- additive prime number theory, cubic residues, 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}
}
@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}
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00929, "Every large non-cube is a prime plus a cube", clopen.solutions.