OPEN
This is open, and cannot be resolved with a finite computation.
Is it true that for every $n\geq 1$ there is a $k$ such that\[n(n+1)\cdots(n+k-1)\mid (n+k)\cdots (n+2k-1)?\]
Asked by Erdős and Straus.
For example when $n=2$ we have $k=5$:\[2\times 3 \times 4 \times 5\times 6 \mid 7 \times 8 \times 9\times 10\times 11.\]and when $n=3$ we have $k=4$:\[3\times 4\times 5\times 6 \mid 7\times 8\times 9\times 10.\]Bhavik Mehta has computed the minimal such $k$ for $1\leq n\leq 18$ (now available as
A375071 on the OEIS).
View the LaTeX source
Additional thanks to: Bhavik Mehta
When referring to this problem, please use the original sources of Erdős. If you wish to acknowledge this website, the recommended citation format is:
T. F. Bloom, Erdős Problem #389, https://www.erdosproblems.com/389, accessed 2026-01-16
