The diagonalization method and Brocard's problem

In this paper, we introduce and develop the method of diagonalization of functions $f:\mathbb{N}\longrightarrow \mathbb{R}$. We apply this method to show that the equations of the form $\Gamma_r(n)+k=m^2$ has a finite number of solutions $n\in \mathbb{N}$ with $n>r$ for any fixed $k,r\in \mathbb{N}$, where $\Gamma_r(n)=n(n-1)\cdots (n-r)$ denotes the $r^{th}$ truncated Gamma function.