Problem Statement
Let $k\geq 2$. Does there exist a prime $p$ and consecutive intervals $I_1,\ldots,I_k$ such that\[\prod_{n\in I_i}n \equiv 1\pmod{p}\]for all $1\leq i\leq k$?
Categories:
Number Theory
Progress
This is problem A15 in Guy's collection [Gu04], where he reports that in a letter in 1979 Erdős observed that\[3\cdot 4\equiv 5\cdot 6\cdot 7\equiv 1\pmod{11},\]establishing the case $k=2$. Makowski [Ma83] found, for $k=3$,\[2\cdot 3\cdot 4\cdot 5\equiv 6\cdot 7\cdot 8\cdot 9\cdot 10\cdot 11\equiv 12\cdot 13\cdot 14\cdot 15\equiv 1\pmod{17}.\]Noll and Simmons asked, more generally, whether there are solutions to $q_1!\equiv\cdots \equiv q_k!\pmod{p}$ for arbitrarily large $k$ (with $q_1<\cdots<q_k$).Source: erdosproblems.com/1056 | Last verified: January 19, 2026