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# A Common Approach to Three Open Problems in Number Theory

Version 1 : Received: 23 March 2023 / Approved: 24 March 2023 / Online: 24 March 2023 (02:00:47 CET)
Version 2 : Received: 28 March 2023 / Approved: 29 March 2023 / Online: 29 March 2023 (03:34:34 CEST)
Version 3 : Received: 14 April 2023 / Approved: 17 April 2023 / Online: 17 April 2023 (04:05:43 CEST)
Version 4 : Received: 18 April 2023 / Approved: 19 April 2023 / Online: 19 April 2023 (05:34:48 CEST)
Version 5 : Received: 26 April 2023 / Approved: 27 April 2023 / Online: 27 April 2023 (04:44:46 CEST)
Version 6 : Received: 10 May 2023 / Approved: 11 May 2023 / Online: 11 May 2023 (04:42:23 CEST)
Version 7 : Received: 17 May 2023 / Approved: 18 May 2023 / Online: 18 May 2023 (08:25:27 CEST)

A peer-reviewed article of this Preprint also exists.

Tyszka, A. A Common Approach to Three Open Problems in Number Theory. Discrete Mathematics Letters 2023, 12, 66–72, doi:10.47443/dml.2023.049. Tyszka, A. A Common Approach to Three Open Problems in Number Theory. Discrete Mathematics Letters 2023, 12, 66–72, doi:10.47443/dml.2023.049.

## Abstract

The following system of equations {x_1 \cdot x_1=x_2, x_2 \cdot x_2=x_3, 2^{2^{x_1}}=x_3, x_4 \cdot x_5=x_2, x_6 \cdot x_7=x_2} has exactly one solution in ({\mathbb N}\{0,1})^7, namely (2,4,16,2,2,2,2). Hypothesis 1 states that if a system of equations S \subseteq {x_i \cdot x_j=x_k: i,j,k \in {1,...,7}} \cup {2^{2^{x_j}}=x_k: j,k \in {1,...,7}} has at most five equations and at most finitely many solutions in ({\mathbb N}\{0,1})^7, then each such solution (x_1,...,x_7) satisfies x_1,...,x_7 \leq 16. Hypothesis 1 implies that there are infinitely many composite numbers of the form 2^{2^{n}}+1. Hypotheses 2 and 3 are of similar kind. Hypothesis 2 implies that if the equation x!+1=y^2 has at most finitely many solutions in positive integers x and y, then each such solution (x,y) belongs to the set {(4,5),(5,11),(7,71)}. Hypothesis 3 implies that if the equation x(x+1)=y! has at most finitely many solutions in positive integers x and y, then each such solution (x,y) belongs to the set {(1,2),(2,3)}. We describe semi-algorithms sem_j (j=1,2,3) that never terminate. For every j \in {1,2,3}, if Hypothesis j is true, then sem_j endlessly prints consecutive positive integers starting from 1. For every j \in {1,2,3}, if Hypothesis j is false, then sem_j prints a finite number (including zero) of consecutive positive integers starting from 1.

## Keywords

Brocard's problem; Brocard-Ramanujan equation x!+1=y^2; composite Fermat numbers; composite numbers of the form 2^(2^n)+1; Erd\"os' equation x(x+1)=y!

## Subject

Computer Science and Mathematics, Algebra and Number Theory

Comment 1
Commenter: Apoloniusz Tyszka
Commenter's Conflict of Interests: Author
Comment: I added Section 1 entitled "Epistemic notions increase the scope of mathematics".
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