Prime solutions of the linear equation ap′ − bp = k
Let $a, b$ be positive integers with $(a, b) = 1$, let $k$ be a nonzero integer, let $P(n)$ be the number of pairs of primes $p, p'$ with $ap' - bp = k$ and $p' < n$, and let $C_2 = \prod_{q \ge 3} \bigl(1 - (q-1)^{-2}\bigr)$, the product running over the odd primes. The source prints the constant below as $2C_2/a$; with $p' < n$ the count of integer solutions is about $n/b$, and $2C_2/b$ is what the heuristic gives.
Problem. Prove that $P(n) = o\bigl(n/(\log n)^2\bigr)$ unless $(k, a) = 1$, $(k, b) = 1$ and exactly one of $k, a, b$ is even; and that if these conditions hold, then $$ P(n) \sim \frac{2 C_2}{b} \frac{n}{(\log n)^2} \prod_{p} \frac{p-1}{p-2}, $$ the product running over the odd primes $p$ dividing $k$, $a$ or $b$.
Source: G. H. Hardy and J. E. Littlewood, Partitio Numerorum III, Acta Math. 44 (1923), Conjecture D.
Imported from doi.org/10.1007/BF02403921.
Classification
- Primary
- NT — Number Theory
- Keywords
- Hardy–Littlewood conjectures, linear equations in primes, prime pairs, 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}
}
@article{bateman1962,
author = {Bateman, Paul T. and Horn, Roger A.},
title = {A heuristic asymptotic formula concerning the distribution of prime numbers},
journal = {Mathematics of Computation},
volume = {16},
pages = {363--367},
year = {1962}
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00922, "Prime solutions of the linear equation ap′ − bp = k", clopen.solutions.