Representations n = ap + bp′ by two primes with fixed coefficients
Let $a, b$ be fixed positive integers with $(a, b) = 1$, let $N(n)$ be the number of representations $n = ap + bp'$ with $p, p'$ prime, and let $C_2 = \prod_{q \ge 3} \bigl(1 - (q-1)^{-2}\bigr)$, the product running over the odd primes. Hardy and Littlewood remark that the first assertion below is trivial.
Problem. Prove that $N(n) = o\bigl(n/(\log n)^2\bigr)$ unless $(n, a) = 1$, $(n, b) = 1$ and exactly one of $n, a, b$ is even; and that if these conditions hold, then $$ N(n) \sim \frac{2 C_2}{ab} \frac{n}{(\log n)^2} \prod_{p} \frac{p-1}{p-2}, $$ the product running over the odd primes $p$ dividing $n$, $a$ or $b$.
Source: G. H. Hardy and J. E. Littlewood, Partitio Numerorum III, Acta Math. 44 (1923), Conjecture C.
Imported from doi.org/10.1007/BF02403921.
Classification
- Primary
- NT — Number Theory
- Keywords
- additive prime number theory, Goldbach's conjecture, Hardy–Littlewood conjectures, singular series
- 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 It will not change if the problem is solved, reclassified or withdrawn.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-00921, "Representations n = ap + bp′ by two primes with fixed coefficients", clopen.solutions.