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.

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.