1. Composite numbers of the form
Let
denote the following system of equations:
The following subsystem of
has exactly one solution in , namely .
Hypothesis 1. If a system of equations has at most five equations and at most finitely many solutions in , then each such solution satisfies .
Lemma 1.([7, p. 109]). For every non-negative integers x and y, if and only if .
Theorem 1. Hypothesis 1 implies that is composite for infinitely many integers greater than 1.
Proof. Assume, on the contrary, that Hypothesis 1 holds and
is composite for at most finitely many integers
greater than 1. Then, the equation
has at most finitely many solutions in
. By Lemma 1, in positive integers greater than 1, the following subsystem of
has at most finitely many solutions in
and expresses that
Since , we get a contradiction. □
Most mathematicians believe that
is composite for every integer
, see [
2].
Open Problem 1.([3], p. 159). Are there infinitely many composite numbers of the form ?
Primes of the form
are called Fermat primes, as Fermat conjectured that every integer of the form
is prime, see [
3, p. 1]. Fermat remarked that
,
,
,
, and
are all prime, see [
3, p. 1].
Open Problem 2.([3, p. 158]). Are there infinitely many prime numbers of the form ?
2. The Brocard-Ramanujan equation
Let
denote the following system of equations:
The following subsystem of
has exactly two solutions in positive integers, namely and .
Hypothesis 2. If a system of equations has at most finitely many solutions in positive integers , then each such solution satisfies .
Lemma 2.
For every positive integers x and y, if and only if
Theorem 2. Hypothesis 2 implies that if the equation has at most finitely many solutions in positive integers and , then each such solution belongs to the set .
Proof. The following system of equations
is a subsystem of
. By Lemma 2, in positive integers, the system
expresses that
or
If the equation has at most finitely many solutions in positive integers and , then has at most finitely many solutions in positive integers and Hypothesis 2 implies that every tuple of positive integers that solves satisfies . Hence, . If , then is a square only for . □
It is conjectured that
is a square only for
, see [
8, p. 297]. A weak form of Szpiro’s conjecture implies that the equation
has only finitely many solutions in positive integers, see [
6].
3. Erdös’ equation
Let
denote the following system of equations:
The following subsystem of
has exactly three solutions in positive integers, namely , , and .
Hypothesis 3. If a system of equations has at most finitely many solutions in positive integers , then each such solution ) satisfies .
Theorem 3. Hypothesis 3 implies that if the equation has at most finitely many solutions in positive integers and , then each such solution belongs to the set .
Proof. The following system of equations
is a subsystem of
. By Lemma 2, in positive integers, the system
expresses that
or
If the equation has at most finitely many solutions in positive integers and , then has at most finitely many solutions in positive integers and Hypothesis 3 implies that every tuple of positive integers that solves satisfies . Hence, . If , then is a product of two consecutive positive integers only for 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
was posed by P. Erdös, see [
1]. F. Luca proved that the
conjecture implies that the equation
has only finitely many solutions in positive integers, see [
4].
4. Hypotheses 2 and 3 cannot be generalized to an arbitrary number of variables
Let
,
, and let
for every integer
. Let
denote the system of equations
. For an integer
, let
denote the following system of equations:
For every positive integer
n, the system
has exactly two solutions in positive integers
, namely
and
. For a positive integer
n, let
denote the following statement:
if a system of equations
has at most finitely many solutions in positive integers , then each such solution satisfies .
Theorem 4. Every factorial Diophantine equation can be algorithmically transformed into an equivalent system of equations of the forms and . 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 2. □
The statement
is dubious. By Theorem 4, 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].
5. Equivalent forms of Hypotheses 1–3
If
, then there are uniquely determined non-negative integers
such that
Definition 1.
For an integer , stands for the smallest system of equations satisfying conditions(1)and(2)
.
(1)If and , then the equation belongs to when it belongs to .
(2)If and , then the equation belongs to when it belongs to .
Lemma 3. .
Proof. It follows from the equality . □
For a positive integer n, let denote the n-th prime number.
Theorem 5.
The following semi-algorithm never terminates.
If Hypothesis 1 is true, then endlessly prints consecutive positive integers starting from 1. If Hypothesis 1 is false, then prints a finite number (including zero) of consecutive positive integers starting from 1.
Proof. It follows from Lemma 3. □
If
, then there are uniquely determined non-negative integers
such that
Definition 2.
For an integer , stands for the smallest system of equations satisfying conditions(3)and(4)
.
(3)If and , then the equation belongs to when it belongs to .
(4)If and , then the equation belongs to when it belongs to .
Lemma 4. .
Proof. It follows from the equality . □
Definition 3.
For an integer , stands for the smallest system of equations satisfying conditions(5)and(6)
.
(5)If and , then the equation belongs to when it belongs to .
(6)If and , then the equation belongs to when it belongs to .
Lemma 5. .
Proof. It follows from the equality . □
Theorem 6.
The following semi-algorithm never terminates.
If Hypothesis 2 is true, then endlessly prints consecutive positive integers starting from 1. If Hypothesis 2 is false, then prints a finite number (including zero) of consecutive positive integers starting from 1.
Proof. It follows from Lemma 4. □
Theorem 7.
The following semi-algorithm never terminates.
If Hypothesis 3 is true, then endlessly prints consecutive positive integers starting from 1. If Hypothesis 3 is false, then prints a finite number (including zero) of consecutive positive integers starting from 1.
Proof. It follows from Lemma 5. □
References
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