Grimm's Conjecture

Conjecture

Let $n$ and $k$ be positive integers.

Let $n 1, n 2, \dots, n k$ be composite numbers.

Then there exists a finite sequence $\left\langle{p_i}\right\rangle$ of $k$ distinct primes such that $p_i$ divides $n i$ for each natural number $i$ such that $1 \le i \le k$.

Source of Name

This entry was named for Carl Albert Grimm.

Sources

• 1969: C.A. Grimm: A conjecture on consecutive composite numbers (Amer. Math. Monthly Vol. 76: pp. 1126 – 1128)  www.jstor.org/stable/2317188