Conjecture 2.2 — Erdős-Woods

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 a positive integer $a$, let $R(a)=\prod_{p\mid a}p$ be its radical, with $R(1)=1$.

Statement recorded in the supplied source.

Does there exist an integer $k\ge1$ such that for every $m,n\ge1$, equality $R(m+i)=R(n+i)$ for all $0\le i<k$ implies $m=n$? [S2]

Short English statement

Can a fixed number of successive prime-divisor sets uniquely determine the starting positive integer?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 8700010 (AMR-086-0010). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Michel Waldschmidt, Open Diophantine Problems, arXiv:math/0312440v2 (24 January 2004), Conjecture 2.2. https://arxiv.org/abs/math/0312440.