On Brocard's problem with Padovan and Perrin numbers

Abstract

The Padovan sequence $\{P_{m}\}_{m\ge 0}$ is a ternary recurrence sequence with companion polynomial $X^{3}-X-1$ and initial conditions $P_{0}=P_{1}=P_{2}=1$. The Perrin sequence $\{R_{m}\}_{m\ge 0}$ is defined by the same companion polynomial as the Padovan sequence, but has initial values $R_{0}=3$, $R_{1}=0$, and $R_{2}=2$. We solve the Brocard-Ramanujan equation $n!+1=x^{2}$, where $n!$ is the factorial of $n$ and $x$ is a Padovan number or a Perrin number. In both cases, we prove that $(n,x)=(4,5)$ is the only solution.