Product of Factorials Equal Another Product of Factorials

Abstract

The Surányi–Hickerson conjecture is a long-standing unsolved problem of Diophantine equations. This conjecture states that all the solutions to \(\ell _1!\cdots \ell _m!=k!\) with \(k-\ell _m\ge 2\) are \((\ell _1,\ldots ,\ell _m;k)=(6,7;10),(3,5,7;10),(2,5,14;16)\) and (2, 3, 3, 7; 9). In this paper, we generalize the Surányi–Hickerson conjecture to \(\ell _1!\cdots \ell _m!=k_1!\cdots k_n!\). We say that a solution \((\ell _1,\ldots ,\ell _m;k_1,\ldots ,k_n)\) is trivial if there exists a pair (i, j) such that \(|\ell _i-k_j|=1\). As in the Surányi–Hickerson conjecture, we give theoretical and computational results. In particular, we suggest that all non-trivial solutions to the equation \(\ell _1!\ell _2=k_1!k_2!\) are \((\ell _1,\ell _2;k_1,k_2)=(7,13;4,15)\), (14, 62; 7, 66) and (22, 54; 18, 57).