A Common Approach to Three Open Problems in Number Theory

1. 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 ${(\mathbb{N}\setminus \{0,1\})}^7$, namely $(2,4,16,2,2,2,2)$.

Hypothesis 1.

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

Lemma 1.([7]). 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 1.

Hypothesis 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 Hypothesis 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 ${(\mathbb{N}\setminus \{0,1\})}^3$. By Lemma 1, in positive integers greater than 1, the following subsystem of $\mathcal{A}$

has at most finitely many solutions in ${(\mathbb{N}\setminus \{0,1\})}^7$ 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 [2]. ([3]). 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 [3]. 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 [3]. ([3]). Are there infinitely many prime numbers of the form $2^{2^n}+1$?

2. An equivalent form of Hypothesis 1

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

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

For every $k\in [{10}^{19},{10}^{20}-1]\cap \mathbb{N}$, we define a system of equations ${\mathcal{S}}_k\subseteq \mathcal{A}$. If $\left\{a\right(0),\dots ,a(19\left)\right\}\cap \{0,8,9\}\ne \varnothing$, then ${\mathcal{S}}_k=\varnothing$. If $\left\{a\right(0),\dots ,a(19\left)\right\}\cap \{0,8,9\}=\varnothing$, then ${\mathcal{S}}_k$ is the smallest system of equations $\mathcal{S}\subseteq \mathcal{A}$ satisfying the following conditions (1a)-(5b).

(1a) If $a\left(3\right)\in \{1,2,3,4\}$, then the equation $2^{2^{x_{a\left(0\right)}}}=x_{a\left(1\right)}$ belongs to $\mathcal{S}$.

(1b) If $a\left(3\right)\in \{5,6,7\}$, then the equation $x_{a\left(0\right)}·x_{a\left(1\right)}=x_{a\left(2\right)}$ belongs to $\mathcal{S}$.

(2a) If $a\left(7\right)\in \{1,2,3,4\}$, then the equation $2^{2^{x_{a\left(4\right)}}}=x_{a\left(5\right)}$ belongs to $\mathcal{S}$.

(2b) If $a\left(7\right)\in \{5,6,7\}$, then the equation $x_{a\left(4\right)}·x_{a\left(5\right)}=x_{a\left(6\right)}$ belongs to $\mathcal{S}$.

(3a) If $a\left(11\right)\in \{1,2,3,4\}$, then the equation $2^{2^{x_{a\left(8\right)}}}=x_{a\left(9\right)}$ belongs to $\mathcal{S}$.

(3b) If $a\left(11\right)\in \{5,6,7\}$, then the equation $x_{a\left(8\right)}·x_{a\left(9\right)}=x_{a\left(10\right)}$ belongs to $\mathcal{S}$.

(4a) If $a\left(15\right)\in \{1,2,3,4\}$, then the equation $2^{2^{x_{a\left(12\right)}}}=x_{a\left(13\right)}$ belongs to $\mathcal{S}$.

(4b) If $a\left(15\right)\in \{5,6,7\}$, then the equation $x_{a\left(12\right)}·x_{a\left(13\right)}=x_{a\left(14\right)}$ belongs to $\mathcal{S}$.

(5a) If $a\left(19\right)\in \{1,2,3,4\}$, then the equation $2^{2^{x_{a\left(16\right)}}}=x_{a\left(17\right)}$ belongs to $\mathcal{S}$.

(5b) If $a\left(19\right)\in \{5,6,7\}$, then the equation $x_{a\left(16\right)}·x_{a\left(17\right)}=x_{a\left(18\right)}$ belongs to $\mathcal{S}$.

Lemma 2.

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

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

Theorem 2.

Hypothesis 1 holds if and only if the following semi-algorithm prints consecutive positive integers starting from 1.

Proof. 

It follows from Lemma 2.    □

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 $(2,2,4,24,24!,(24!)!)$.

Hypothesis 2.

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 3.

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

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

Theorem 3.

Hypothesis 2 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 3, 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 Hypothesis 2 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]. A weak form of Szpiro’s conjecture implies that the equation $x!+1=y^2$ has only finitely many solutions in positive integers, see [6].

4. Erdös’ equation $x(x+1)=y!$

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 $(2,2,3,6,720,720!)$.

Hypothesis 3.

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 4.

Hypothesis 4 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 3, 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 Hypothesis 3 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 [1]. F. Luca proved that the $abc$ conjecture implies that the equation $x(x+1)=y!$ has only finitely many solutions in positive integers, see [4].

5. There is no hope for a hypothesis that is similar to Hypothesis 2 or 3 and holds for 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 ${\mathcal{U}}_1$ denote the system of equations $\{x_1!=x_1$. For an integer $n⩾2$, let ${\mathcal{U}}_n$ denote the following system of equations:

For every positive integer n, the system ${\mathcal{U}}_n$ 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 5.

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 [7].

Proof. 

It follows from Lemmas 2–4 in [7] and Lemma 3. □

The statement $\forall n\in \mathbb{N}\setminus \left\{0\right\}{\Psi }_n$ is dubious. By Theorem 5, 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 [5].