Newman's Conjecture

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

For $n\geq1$, let $p(n)$ be the number of integer partitions of $n$, that is, the number of finite nonincreasing sequences of positive integers with sum $n$. Set $p(0)=1$. For integers $m\geq1$ and $a$, a unit residue means $\gcd(a,m)=1$.

Statement recorded in the supplied source.

For every integer $m\geq1$ and every integer $a$ with $\gcd(a,m)=1$, is the set \[ \{n\in\mathbb N:p(n)\equiv a\pmod m\} \] infinite? [S2] The usual Newman conjecture asks the same question for every residue $a$, including nonunits. The recorded unit-residue question is the weaker restriction appearing in this catalogue's background. [S2]

Short English statement

Does every residue coprime to its modulus occur infinitely often among partition numbers?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 1245 (NT-038). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Dohoon Choi and Youngmin Lee, Newman's conjecture for the partition function modulo integers with at least two distinct prime divisors, arXiv:2212.00636v2 (2025), Section 1, Conjecture 1.1. https://arxiv.org/pdf/2212.00636.