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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.

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

Copyright:
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Commenter: Apoloniusz Tyszka

Commenter's Conflict of Interests: Author