On the Brocard–Ramanujan problem and generalizations

Andrzej Dąbrowski¹

¹ Institute of Mathematics University of Szczecin Wielkopolska 15 70-451 Szczecin, Poland

Colloquium Mathematicum, Tome 126 (2012) no. 1, pp. 105-110

Résumé

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).

Détail

DOI : 10.4064/cm126-1-7

Keywords: denote ith prime conjecture there precisely solutions equation alpha cdots alpha positive integers alpha ldots alpha conjecture implies explicit description set solutions brocard ramanujan equation propose another variant brocard ramanujan problem describe set solutions non negative integers equation fixed