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.

Equations

$$f(n)+k=m^2.$$

$${\mathbb{B}}_k(f):=\left\{m^2\in \mathbb{N}|f(n)+k=m^2\right\}$$

$$|{\mathcal{D}}_k(f,s)|:=\#\{n\le s|f(n)+k=m^2\}.$$

$${\mathbb{T}}_f(s,k):=\sum _{\begin{array}{c}n\le s\\ n\in {\mathcal{D}}_k(f)\end{array}}f(n).$$

$${\mathbb{T}}_f(s,k):$$

$$=\underset{1^{-}}{\overset{s}{\int }}f(t)d|{\mathcal{D}}_k(f,t)|$$

$$=f(s)|{\mathcal{D}}_k(f,s)|-\underset{1}{\overset{s}{\int }}{f}^{\prime }(t)|{\mathcal{D}}_k(f,t)|dt.$$

$$\displaystyle\frac{\sqrt{s}}{f(s)}\bigg{(}\sqrt{\int\limits_{1}^{s}|f^{\prime}(t)|^{2}dt}\bigg{)}<1$$

$$\displaystyle|\mathcal{D}_{k}(f,s)|\ll\bigg{(}\frac{1}{f(s)}\sum\limits_{n\leq s}f(n)\bigg{)}\bigg{(}1-\frac{\sqrt{s}}{f(s)}\sqrt{\int\limits_{1}^{s}|f^{\prime}(t)|^{2}dt}\bigg{)}^{-1}.$$

$$|{\mathcal{D}}_k(f,s)|$$

$$\displaystyle\bigg{(}\int\limits_{1}^{s}|\mathcal{D}_{k}(f,t)|^{2}dt\bigg{)}^{\frac{1}{2}}\ll|\mathcal{D}_{k}(f,s)|\sqrt{s}$$

$$\displaystyle\mathcal{D}_{k}(f,s)\bigg{(}1-\frac{\sqrt{s}}{f(s)}\bigg{(}\sqrt{\int\limits_{1}^{s}|f^{\prime}(t)|^{2}dt}\bigg{)}<\frac{1}{f(s)}\sum\limits_{n\leq s}f(n)$$

$$\displaystyle\frac{\sqrt{s}}{f(s)}\bigg{(}\sqrt{\int\limits_{1}^{s}|f^{\prime}(t)|^{2}dt}\bigg{)}<1$$

$$\displaystyle\lim\limits_{s\longrightarrow\infty}\bigg{(}\frac{1}{f(s)}\sum\limits_{n\leq s}f(n)\bigg{)}\bigg{(}1-\frac{\sqrt{s}}{f(s)}\sqrt{\int\limits_{1}^{s}|f^{\prime}(t)|^{2}dt}\bigg{)}^{-1}<\infty$$

$$|{\mathcal{D}}_k(f,s)|:=\#\{n\le s|f(n)+k=m^2\}$$

$${\mathcal{D}}_k(f)\cap {\mathbb{N}}_s:={\mathcal{D}}_k(f,s).$$

$$\#\{n\le s|f(n)+k=m^2\}$$

$${\Gamma }_r(n):=\{\begin{array}{l}n(n-1)\cdots (n-r)\mathbf{if}n>r\\ 0\mathbf{otherwise}.\end{array}$$

$${\Gamma }_r(s)=s(s-1)\cdots (s-r).$$

$${\Gamma }_r(s-1)=(s-1)(s-2)\cdots (s-1-(r-1))$$

$${\Gamma }_r(s-2)=(s-2)\cdots (s-2-(r-2))$$

$$\frac{1}{{\Gamma }_r(s)}\sqrt{\underset{1}{\overset{s}{\int }}|{\Gamma }_r^{\prime }(t){|}^{2}dt}\asymp \frac{1}{s^{\frac{1}{2}}}.$$

$$\underset{1}{\overset{s}{\int }}|{\Gamma }_r^{\prime }(t){|}^{2}dt\asymp s^{2r+1}$$

$$\underset{s⟶\infty }{\lim }\frac{1}{{\Gamma }_r(s)}\sum _{n\le s}{\Gamma }_r(n)<\infty$$

$$\underset{s⟶\infty }{\lim }\frac{\sqrt{s}}{{\Gamma }_r(s)}\sqrt{\underset{1}{\overset{s}{\int }}|{\Gamma }_r^{\prime }(t){|}^{2}dt}<\infty$$

$$\frac{\sqrt{s}}{{\Gamma }_r(s)}\sqrt{\underset{1}{\overset{s}{\int }}|{\Gamma }_r^{\prime }(t){|}^{2}dt}<1$$

Full text

1. Introduction and problem statement

Brocard’s problem is an unsolved problem which - roughly speaking - asks if the set of integers whose factorials are unit left translate of a square is either finite or infinite. As the origin of the problem suggests, it was first formulated by the French mathematician Henri Brocard around 1876 and 1885 and subsequently rediscovered by the Indian mathematician Srinivasa Ramanujan in 1913. More formally the problem states

Problem 1.1*.*

Does the equation $n!+1=m^{2}$ has integer solutions other than $4,5,7$?

It is widely believed any other solutions to Brocard’s equation - if they exist - must be finite. In fact, Paul Erdős conjectured that no further solution to Brocard’s equation exists. It has also been verified computationally for numbers up to $10^{9}$ for a possible solution to the equation and within this threshold no further solution to Brocard’s problem has been found [4]. Albeit the problem remains unsolved, a good number of theoretical progress has been made. The first major progress has been made by M. Overholt, who showed that there are only a finite number of integer solutions to the Brocard equation $n!+1=m^{2}$ by assuming the ABC conjecture [1]. These result has also been extended to equations with arbitrary shifts of the form $n!+A=m^{2}$ [2]. A further extension has been made in [3] where it is shown that the equation $n!=P(x)$, where $P(x)$ is a polynomial of degree at least two, also has a finite number of solution by assuming the ABC conjecture. It is important to recognize that Brocard’s equation is equivalent to the equation $\Gamma(n+1)+k=m^{2}$ for $n\in\mathbb{N}$, where $\Gamma$ denotes the Gamma function.

In this paper we apply the method of diagonalization of functions to show that the equations of the form $\Gamma_{r}(n)+k=m^{2}$ has a finite number of solutions for $n\in\mathbb{N}$, where $\Gamma_{r}$ denotes the $r^{th}$ truncated Gamma function.

2. The notion of diagonalization

In this section we introduce and study the notion of diagonalization of a function. We study this notion together with associated statistics and explore some applications.

Definition 2.1**.**

Let $f:\mathbb{N}\longrightarrow\mathbb{R}$. Then we say $f$ is $k$ - step diagonalizable at the spot $n\in\mathbb{N}$ if there exists some $m\in\mathbb{N}$ such that

[TABLE]

We call the set of all spots $n\in\mathbb{N}$ such that $f$ is $k$ - step diagonalizable the $k^{th}$ - step diagonal of $f$ and denote by $\mathcal{D}_{k}(f)$. We call the set of all truncated spots $\mathcal{D}_{k}(f)\cap\mathbb{N}_{s}:=\mathcal{D}_{k}(f,s)$ the $s^{th}$ scale diagonal. We call the set of all squares

[TABLE]

the $k^{th}$-step diagonal squares. We write the length of this diagonal as

[TABLE]

It is easy to see that $|\mathcal{D}_{k}(f,s)|<s$.

2.1. The $s$-level trace of the diagonal

In this section we introduce the notion of the trace of the diagonal. We launch and examine the following languages.

Definition 2.2**.**

By the $s^{th}$ level trace of the diagonal $\mathcal{D}_{k}(f)$, denoted $\mathbb{T}_{f}(s,k)$, we mean the partial sum

[TABLE]

Let us suppose that $f$ is a function with continuous derivative on $[1,s]$ for $s\geq 1$ with $s\in\mathbb{R}$, then by applying the Stieltjes integration by parts, we can write the $s^{th}$ level trace of the diagonal in the form

[TABLE]

Theorem 2.3** (Diagonal inequality).**

Let $f$ be a function with continuous derivative on $[1,s]$ for $s\geq 1$ with $s\in\mathbb{R}$. If

[TABLE]

for all $s\geq 1$ then the inequality holds

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Proof.

By appealing to the ensuing discussion, we obtain the upper bound

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so that by appealing to the Cauchy-Schwartz inequality, we obtain further the upper bound

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so that by exploiting the bound

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rearranging terms, appealing to the conditions we obtain the inequality

[TABLE]

and the claimed inequality holds immediately. ∎

Theorem 2.3 supplies a useful inequality to study Brocard’s problem, which ask whether there exists a finite number of solutions to the equation $n!+1=m^{2}$. The current development can be leveraged to study a much more general version of the problem. By applying the Diagonal inequality we obtain

Proposition 2.1** (The Diagonal method).**

Let

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for all $s\geq 1$. If

[TABLE]

then the equation $f(n)+k=m^{2}$ has only a finite number of solutions in $\mathbb{N}$ for a fixed $k\in\mathbb{N}$.

Proof.

Appealing to Theorem 2.3, it follows under the requirements that $|\mathcal{D}_{k}(f)|<\infty$, and the claim follows immediately since

[TABLE]

with

[TABLE]

∎

Remark 2.4*.*

The upper bound derived in Proposition 2.3 supplies a somewhat useful tool to study the size of the quantity

[TABLE]

and in particular Brocard’s problem which asks if the set of integers whose factorials are unit left translate of a square is either an infinite set or a finite set. It is worth noting that the upper bounds we have derived do not depend on the size of the shift but on the underlying function. This uniformity does suggest the actual size of the quantity

[TABLE]

will mostly be influenced by the function under consideration. In various circumstances the ease with which to verify the underlying conditions will inform the category of bounds to exploit. Now we apply the Diagonal method to study Brocard’s problem.

2.2. Analogous Brocard problem

In this section we apply the diagonalization method to an analogue of Brocard’s problem. We first introduce a certain truncation of the Euler Gamma function.

Definition 2.5** (The $r^{th}$ truncated Gamma function).**

Let $r\in\mathbb{N}$ be fixed. Then by the $r^{th}$ truncated Gamma function $\Gamma_{r}$, we mean the function

[TABLE]

It follows the Definition of the $r^{th}$ truncated Gamma function that for $s>r$ then

[TABLE]

It follows similarly that

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since $s-1>r-1$ and

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since $s-2>r-2$ and so on. The $r^{th}$ truncated Gamma function for smaller arguments can be obtained in a similar manner.

Lemma 2.6**.**

For all $s>r$, we have

[TABLE]

Proof.

It follows naturally from the definition of the $r^{th}$ truncated Gamma function that $\Gamma_{r}(s)\asymp s^{r+1}$ so that

[TABLE]

and the claimed upper bound is an easy consequence. ∎

Theorem 2.7** (Variational Brocard).**

The equation $\Gamma_{r}(s)+k=m^{2}$ has finite number of solutions $s\in\mathbb{N}$ with $s>r$ for a fixed $k,r\in\mathbb{N}$, where $\Gamma_{r}$ is the $r^{th}$ truncated Euler Gamma function.

Proof.

It suffices to check that

[TABLE]

and that

[TABLE]

with

[TABLE]

so that $|\mathcal{D}_{k}(\Gamma_{r})|<\infty$ for any fixed $k\in\mathbb{N}$. ∎