Conjecture 1.7

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 $N$, let $\omega(N)$ be the number of its distinct prime divisors. Consider $n,k\in\mathbb N$, so the consecutive block $n+1,\ldots,n+k$ lies above one.

Statement recorded in the supplied source.

If the interval $[n+1,n+k]$ contains no prime, must \[\omega\left((n+1)\cdots(n+k)\right)\ge k?\] [S2]

Short English statement

Must a consecutive block of composite numbers have at least as many different prime factors collectively as it has terms?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 8700006 (AMR-086-0006). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Michel Waldschmidt, Open Diophantine Problems, arXiv:math/0312440v2 (24 January 2004), Conjecture 1.7. https://arxiv.org/abs/math/0312440.