# The diagonalization method and Brocard's problem

## TL;DR

This paper introduces a diagonalization technique for functions and applies it to demonstrate that certain equations related to Brocard's problem have finitely many solutions for fixed parameters.

## Contribution

The paper develops a new diagonalization method for functions and uses it to analyze the finiteness of solutions to specific equations connected to Brocard's problem.

## Key findings

- Equations of the form Γ_r(n)+k=m^2 have finitely many solutions for fixed k,r.
- The diagonalization method provides a novel approach to studying these equations.
- Application to Brocard's problem offers new insights into the solution structure.

## Abstract

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.