Representations n = am² + bm + p by a quadratic and a prime
Let $a, b$ be integers with $a > 0$, let $N(n)$ be the number of representations $n = am^2 + bm + p$ with $m \ge 0$ an integer and $p$ prime, let $\varepsilon = 1$ if $a + b$ is odd and $\varepsilon = 2$ if $a + b$ is even, and let $\bigl(\frac{\cdot}{q}\bigr)$ be the Legendre symbol.
Problem. Prove that $N(n) = o\bigl(\sqrt n/\log n\bigr)$ if $n, a, b$ have a common factor, or if $n$ and $a + b$ are both even, or if $b^2 + 4an$ is a square; and that in all other cases $$ N(n) \sim \frac{\varepsilon}{\sqrt a} \frac{\sqrt n}{\log n} \prod_{p} \frac{p}{p-1} \prod_{q \ge 3,\ q \nmid a} \left(1 - \frac{1}{q-1} \left(\frac{b^2 + 4an}{q}\right)\right), $$ the first product running over the odd primes $p$ dividing both $a$ and $b$, the second over the odd primes $q$ not dividing $a$.
Source: G. H. Hardy and J. E. Littlewood, Partitio Numerorum III, Acta Math. 44 (1923), Conjecture G.
Imported from doi.org/10.1007/BF02403921.
Classification
- Primary
- NT — Number Theory
- Keywords
- additive prime number theory, Hardy–Littlewood conjectures, quadratic 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
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OP-00925, "Representations n = am² + bm + p by a quadratic and a prime", clopen.solutions.