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.