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Abstract

Let $p_i$ denote the $i$th prime. We conjecture that there are precisely $28$ solutions to the equation $n^2-1=p_1^{\alpha_1}\cdots p_k^{\alpha_k}$ in positive integers $n$ and $\alpha_1$,\ldots ,$\alpha_k$. This conjecture implies an explicit description of the set of solutions to the Brocard–Ramanujan equation. We also propose another variant of the Brocard–Ramanujan problem: describe the set of solutions in non-negative integers of the equation $n!+A=x_1^2+x_2^2+x_3^2$ ($A$ fixed).