MacEachen 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.

Let $p_1=2,p_2=3,\ldots$ be the primes in increasing order and $P_j=\prod_{i=1}^j p_i$ their primorials. Let $\mathcal S=\{\prod_{j\in J}P_j:J\subseteq\mathbb N\text{ finite}\}$, with the empty product equal to $1$; each primorial may occur at most once. This is OEIS A129912. Two integers are adjacent when their absolute difference is $1$. The author-confirmed distance restriction is included as in the OEIS comment.

Statement recorded in the supplied source.

For every odd prime $p$, does there exist $s\in\mathcal S$ such that either $|p-s|=1$, or $|p-s|$ is a prime $q<p$? [S2] [S3]

Short English statement

Is every odd prime within distance one, or within a smaller prime distance, of a product of distinct primorials?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 3345 (OPG-671). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Open Problem Garden, MacEachen Conjecture, node 671, original problem page. Original question and full discussion. https://www.openproblemgarden.org/op/maceachen_conjecture.
• S3 Bill McEachen, OEIS A129912, Numbers that are products of distinct primorial numbers, author comment of 3 June 2010 (clarified 26 February 2019), official OEIS data repository. https://raw.githubusercontent.com/oeis/oeisdata/main/seq/A129/A129912.seq.