Prime pairs m² + 1, m² + 3
Let $Q(n)$ be the number of pairs of primes $m^2 + 1,\ m^2 + 3$ less than $n$. For a prime $q \ge 5$, let $\nu(q) = 0$, $2$ or $4$ according as neither, one or both of $-1$ and $-3$ are quadratic residues of $q$. Hardy and Littlewood pose this as the one problem they considered that combines prime pairs with primes of the form $m^2 + 1$; they report 57 such pairs below $9\,000\,000$, against $48.9$ from the formula below (in an equivalent integral form).
Problem. Prove that there are infinitely many prime pairs of the form $m^2 + 1,\ m^2 + 3$, and that $$ Q(n) \sim \frac{3 \sqrt n}{(\log n)^2} \prod_{q \ge 5} \frac{q\,(q - \nu(q))}{(q-1)^2} \qquad (n \to \infty). $$
Source: G. H. Hardy and J. E. Littlewood, Partitio Numerorum III, Acta Math. 44 (1923), Conjecture P.
Imported from doi.org/10.1007/BF02403921.
Classification
- Primary
- NT — Number Theory
- Keywords
- Hardy–Littlewood conjectures, prime pairs, primes represented by 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
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00932, "Prime pairs m² + 1, m² + 3", clopen.solutions.