A Common Approach to Three Open Problems in Number Theory

1. Epistemic Notions Increase the Scope of Mathematics

Nicolas D. Goodman observed that epistemic notions increase the scope of mathematics, see [1]. For many finite sets $\mathcal{X}\subseteq {\mathbb{N}}^m$, we know an algorithm that decides $\mathcal{X}$ although no known algorithm computes a positive integer n satisfying $\mathcal{X}\subseteq {[0,n]}^m$. This holds as for many Diophantine equations the number of rational solutions is finite by applying Faltings’ theorem. Faltings’ theorem tell us that certain curves have at most finitely many rational points, but no known proof gives any bound on the sizes of the numerators and denominators of the coordinates of those points.

In Section 2, Section 3 and Section 4, our knowledge (including conjectures) about the set $\mathcal{X}$ is different. The considerations in Section 2 imply the existence of the set ${\mathcal{X}}_2\subseteq {(\mathbb{N}\backslash \{0,1\})}^7$ whose finiteness/infiniteness is unknown although we conjecture that $\mathrm{card}\left({\mathcal{X}}_2\right)<\omega \Rightarrow {\mathcal{X}}_2\subseteq {[2,16]}^7$. The considerations in Section 3 imply the existence of the set ${\mathcal{X}}_3\subseteq {(\mathbb{N}\backslash \left\{0\right\})}^6$ whose finiteness/infiniteness is unknown although we conjecture that $\mathrm{card}\left({\mathcal{X}}_3\right)<\omega \Rightarrow {\mathcal{X}}_3\subseteq {[1,(24!)!]}^6$. The considerations in Section 4 imply the existence of the set ${\mathcal{X}}_4\subseteq {(\mathbb{N}\backslash \left\{0\right\})}^6$ whose finiteness/infiniteness is unknown although we conjecture that $\mathrm{card}\left({\mathcal{X}}_4\right)<\omega \Rightarrow {\mathcal{X}}_4\subseteq {[1,720!]}^6$. For every $j\in \{2,3,4\}$, we know an algorithm that decides the set ${\mathcal{X}}_j$.

2. Composite Numbers of the Form $2^{2^n}+1$

Let $\mathcal{A}$ denote the following system of equations:

$$\left\{x_i·x_j=x_k:i,j,k\in \{1,\dots ,7\}\right\}\cup \left\{2^{2^{x_j}}=x_k:j,k\in \{1,\dots ,7\}\right\}$$

The following subsystem of $\mathcal{A}$

has exactly one solution in ${(\mathbb{N}\backslash \{0,1\})}^7$, namely $(2,4,16,2,2,2,2)$.

Conjecture 1.

If a system of equations $\mathcal{S}\subseteq \mathcal{A}$ has at most five equations and at most finitely many solutions in ${(\mathbb{N}\backslash \{0,1\})}^7$, then each such solution $(x_1,\dots ,x_7)$ satisfies $x_1,\dots ,x_7⩽16$.

Lemma 1.

[[2] (p. 109)]. For every non-negative integers x and y, $x+1=y$ if and only if $2^{2^x}·2^{2^x}=2^{2^y}$.

Theorem 2.

Conjecture 1 implies that $2^{2^{x_1}}+1$ is composite for infinitely many integers $x_1$ greater than 1.

Proof. 

Assume, on the contrary, that Conjecture 1 holds and $2^{2^{x_1}}+1$ is composite for at most finitely many integers $x_1$ greater than 1. Then, the equation

$$x_2·x_3=2^{2^{x_1}}+1$$

has at most finitely many solutions in ${(\mathbb{N}\backslash \{0,1\})}^3$. By Lemma 1, in positive integers greater than 1, the following subsystem of $\mathcal{A}$

has at most finitely many solutions in ${(\mathbb{N}\backslash \{0,1\})}^7$ and expresses that

$$\left\{\begin{array}{ccc}\hfill x_2·x_3& =& 2^{2^{x_1}}+1\hfill \\ \hfill x_4& =& 2^{2^{x_1}}+1\hfill \\ \hfill x_5& =& 2^{2^{x_1}}\hfill \\ \hfill x_6& =& 2^{2^{2^{2^{x_1}}}}\hfill \\ \hfill x_7& =& 2^{2^{2^{2^{x_1}}+1}}\hfill \end{array}\right.$$

Since $641·6700417=2^{2^5}+1>16$, we get a contradiction.    □

Most mathematicians believe that $2^{2^n}+1$ is composite for every integer $n⩾5$, see [3, p. 23].

Open Problem 1

([4] (p. 159)). Are there infinitely many composite numbers of the form $2^{2^n}+1$?

Primes of the form $2^{2^n}+1$ are called Fermat primes, as Fermat conjectured that every integer of the form $2^{2^n}+1$ is prime, see [4] (p. 1). Fermat remarked that $2^{2^0}+1=3$, $2^{2^1}+1=5$, $2^{2^2}+1=17$, $2^{2^3}+1=257$, and $2^{2^4}+1=65537$ are all prime, see [4, p. 1].

Open Problem 2

([4] (p. 158)). Are there infinitely many prime numbers of the form $2^{2^n}+1$?

3. The Brocard-Ramanujan equation $x!+1=y^2$

Let $\mathcal{B}$ denote the following system of equations:

$$\{x_i·x_j=x_k:i,j,k\in \{1,\dots ,6\}\}\cup \{x_j!=x_k:(j,k\in \{1,\dots ,6\})\wedge (j\ne k)\}$$

The following subsystem of $\mathcal{B}$

has exactly two solutions in positive integers, namely $(1,\dots ,1)$ and $(2,2,4,24,24!,(24!)!)$.

Conjecture 3.

If a system of equations $\mathcal{S}\subseteq \mathcal{B}$ has at most finitely many solutions in positive integers $x_1,\dots ,x_6$, then each such solution $(x_1,\dots ,x_6)$ satisfies $x_1,\dots ,x_6⩽(24!)!$.

Lemma 2.

For every positive integers x and y, $x!·y=y!$ if and only if

$$(x+1=y)\vee (x=y=1)$$

Theorem 4.

Conjecture 3 implies that if the equation $x_1!+1=x_2^2$ has at most finitely many solutions in positive integers $x_1$ and $x_2$, then each such solution $(x_1,x_2)$ belongs to the set $\left\{\right(4,5),(5,11),(7,71\left)\right\}$.

Proof. 

The following system of equations ${\mathcal{B}}_1$

is a subsystem of $\mathcal{B}$. By Lemma 2, in positive integers, the system ${\mathcal{B}}_1$ expresses that $x_1=\dots =x_6=1$ or

$$\left\{\begin{array}{ccc}\hfill x_1!+1& =& x_2^2\hfill \\ \hfill x_3& =& x_1!\hfill \\ \hfill x_4& =& (x_1!)!\hfill \\ \hfill x_5& =& x_1!+1\hfill \\ \hfill x_6& =& (x_1!+1)!\hfill \end{array}\right.$$

If the equation $x_1!+1=x_2^2$ has at most finitely many solutions in positive integers $x_1$ and $x_2$, then ${\mathcal{B}}_1$ has at most finitely many solutions in positive integers $x_1,\dots ,x_6$ and Conjecture 3 implies that every tuple $(x_1,\dots ,x_6)$ of positive integers that solves ${\mathcal{B}}_1$ satisfies $(x_1!+1)!=x_6⩽(24!)!$. Hence, $x_1\in \{1,\dots ,23\}$. If $x_1\in \{1,\dots ,23\}$, then $x_1!+1$ is a square only for $x_1\in \{4,5,7\}$.    □

It is conjectured that $x!+1$ is a square only for $x\in \{4,5,7\}$, see [8] (p. 297). A weak form of Szpiro’s conjecture implies that the equation $x!+1=y^2$ has only finitely many solutions in positive integers, see [9].

4. Erdös’ Equation $\mathbf{x}(\mathbf{x}+\mathbf{1})=\mathbf{y}!$

Let $\mathcal{C}$ denote the following system of equations:

$$\{x_i·x_j=x_k:(i,j,k\in \{1,\dots ,6\})\wedge (i\ne j)\}\cup \{x_j!=x_k:(j,k\in \{1,\dots ,6\})\wedge (j\ne k)\}$$

The following subsystem of $\mathcal{C}$

has exactly three solutions in positive integers, namely $(1,\dots ,1)$, $(1,1,2,2,2,2)$, and $(2,2,3,6,720,720!)$.

Conjecture 5.

If a system of equations $\mathcal{S}\subseteq \mathcal{C}$ has at most finitely many solutions in positive integers $x_1,\dots ,x_6$, then each such solution $(x_1,\dots ,x_6$) satisfies $x_1,\dots ,x_6⩽720!$.

Theorem 6.

Conjecture 5 implies that if the equation $x_1(x_1+1)=x_2!$ has at most finitely many solutions in positive integers $x_1$ and $x_2$, then each such solution $(x_1,x_2)$ belongs to the set $\left\{\right(1,2),(2,3\left)\right\}$.

Proof. 

The following system of equations ${\mathcal{C}}_1$

is a subsystem of $\mathcal{C}$. By Lemma 2, in positive integers, the system ${\mathcal{C}}_1$ expresses that $x_1=\dots =x_6=1$ or

$$\left\{\begin{array}{ccc}\hfill x_1·(x_1+1)& =& x_2!\hfill \\ \hfill x_3& =& x_1·(x_1+1)\hfill \\ \hfill x_4& =& x_1!\hfill \\ \hfill x_5& =& x_1+1\hfill \\ \hfill x_6& =& (x_1+1)!\hfill \end{array}\right.$$

If the equation $x_1(x_1+1)=x_2!$ has at most finitely many solutions in positive integers $x_1$ and $x_2$, then ${\mathcal{C}}_1$ has at most finitely many solutions in positive integers $x_1,\dots ,x_6$ and Conjecture 5 implies that every tuple $(x_1,\dots ,x_6)$ of positive integers that solves ${\mathcal{C}}_1$ satisfies $x_2!=x_3⩽720!$. Hence, $x_2\in \{1,\dots ,720\}$. If $x_2\in \{1,\dots ,720\}$, then $x_2!$ is a product of two consecutive positive integers only for $x_2\in \{2,3\}$ because the following MuPAD program

for x2 from 1 to 720 do

x1:=round(sqrt(x2!+(1/4))-(1/2)):

if x1*(x1+1)=x2! then print(x2) end_if:

end_for:

returns 2 and 3.    □

The question of solving the equation $x(x+1)=y!$ was posed by P. Erdös, see [5]. F. Luca proved that the $abc$ conjecture implies that the equation $x(x+1)=y!$ has only finitely many solutions in positive integers, see [6].

5. Conjectures 3 and 5 Cannot Be Generalized to An Arbitrary Number of Variables

Let $f\left(1\right)=2$, $f\left(2\right)=4$, and let $f(n+1)=f\left(n\right)!$ for every integer $n⩾2$. Let ${\mathcal{W}}_1$ denote the system of equations $\{x_1!=x_1$. For an integer $n⩾2$, let ${\mathcal{W}}_n$ denote the following system of equations:

For every positive integer n, the system ${\mathcal{W}}_n$ has exactly two solutions in positive integers $x_1,\dots ,x_n$, namely $(1,\dots ,1)$ and $\left(f\right(1),\dots ,f(n\left)\right)$. For a positive integer n, let ${\Psi }_n$ denote the following statement: if a system of equations

$$\mathcal{S}\subseteq \{x_i·x_j=x_k:i,j,k\in \{1,\dots ,n\}\}\cup \{x_j!=x_k:j,k\in \{1,\dots ,n\}\}$$

has at most finitely many solutions in positive integers $x_1,\dots ,x_n$, then each such solution $(x_1,\dots ,x_n)$ satisfies $x_1,\dots ,x_n⩽f\left(n\right)$.

Theorem 7.

Every factorial Diophantine equation can be algorithmically transformed into an equivalent system of equations of the forms $x_i·x_j=x_k$ and $x_j!=x_k$. It means that this system of equations satisfies a modified version of Lemma 4 in [2].

Proof. 

It follows from Lemmas 2–4 in [2] and Lemma 2.    □

The statement $\forall n\in \mathbb{N}\backslash \left\{0\right\}{\Psi }_n$ is dubious. By Theorem 7, this statement implies that there is an algorithm which takes as input a factorial Diophantine equation and returns an integer which is greater than the solutions in positive integers, if these solutions form a finite set. This conclusion is strange because properties of factorial Diophantine equations are similar to properties of exponential Diophantine equations and a computable upper bound on non-negative integer solutions does not exist for exponential Diophantine equations with a finite number of solutions, see [7].

6. Equivalent Forms of Conjectures 1–5

If $k\in [{10}^{19},{10}^{20}-1]\cap \mathbb{N}$, then there are uniquely determined non-negative integers $a\left(0\right),\dots ,a\left(19\right)\in \{0,\dots ,9\}$ such that

$$\left(a\left(19\right)⩾1\right)\wedge (k=a\left(19\right)·{10}^{19}+a\left(18\right)·{10}^{18}+\dots +a\left(1\right)·{10}^1+a\left(0\right)·{10}^0)$$

Definition 1.

For an integer $k\in [{10}^{19},{10}^{20}-1]$, ${\mathcal{S}}_k$ stands for the smallest system of equations $\mathcal{S}$ satisfying conditions(1)and(2).

(1)If $i\in \{0,4,8,16\}$ and $a\left(i\right)$ is even, then the equation $x_{a(i+1)}·x_{a(i+2)}=x_{a(i+3)}$ belongs to $\mathcal{S}$ when it belongs to $\mathcal{A}$.

(2)If $i\in \{0,4,8,16\}$ and $a\left(i\right)$ is odd, then the equation $2^{2^{x_{a(i+1)}}}=x_{a(i+2)}$ belongs to $\mathcal{S}$ when it belongs to $\mathcal{A}$.

Lemma 3.

$\{{\mathcal{S}}_k:k\in [{10}^{19},{10}^{20}-1]\cap \mathbb{N}=\{\mathcal{S}:(\mathcal{S}\subseteq \mathcal{A})\wedge (\mathrm{card}\left(\mathcal{S}\right)⩽5)\}$.

Proof. 

It follows from the equality $5·4=20$. □

For a positive integer n, let $p_n$ denote the n-th prime number.

Theorem 8.

The following semi-algorithm ${\mathtt{sem}}_1$ never terminates.

If Conjecture 1 is true, then ${\mathtt{sem}}_1$ endlessly prints consecutive positive integers starting from 1. If Conjecture 1 is false, then ${\mathtt{sem}}_1$ prints a finite number (including zero) of consecutive positive integers starting from 1.

Proof. 

It follows from Lemma 3. □

Theorem 9.

The following semi-algorithm ${\mathtt{sem}}_2$ never terminates.

If Conjecture 3 is true, then ${\mathtt{sem}}_2$ endlessly prints consecutive positive integers starting from 1. If Conjecture 3 is false, then ${\mathtt{sem}}_2$ prints a finite number (including zero) of consecutive positive integers starting from 1.

Theorem 10.

The following semi-algorithm ${\mathtt{sem}}_3$ never terminates.

If Conjecture 5 is true, then ${\mathtt{sem}}_3$ endlessly prints consecutive positive integers starting from 1. If Conjecture 5 is false, then ${\mathtt{sem}}_3$ prints a finite number (including zero) of consecutive positive integers starting from 1.