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 $q_1<\cdots<q_k$).

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This page was last edited 29 September 2025.