Conjecture 2.3 — Erdős–Dressler

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.

Use $R(a)=\prod_{p\mid a}p$ for the radical of a positive integer.

Statement recorded in the supplied source.

For every $1\le a<b$ with $R(a)=R(b)$, must there be a prime $p$ satisfying $a<p<b$? [S2]

Short English statement

Must two different positive integers with the same prime divisors have a prime strictly between them?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 8700011 (AMR-086-0011). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Michel Waldschmidt, Open Diophantine Problems, arXiv:math/0312440v2 (24 January 2004), Conjecture 2.3. https://arxiv.org/abs/math/0312440.