Product of Factorials Equal Another Product of Factorials

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