We discuss and give some insights on the Collatz conjecture, known as 3N + 1, and Kurepa’s hypothesis on the left factorial. First, the Collatz conjecture is considered and the density of values is compared to Planck’s black body radiation in physics, showing a remarkable agreement between the two. We also briefly discuss a generalization of Collatz conjecture for a generic sequence qN +1 by means of numerical analysis. Then, we give a brief historical excursus and prove in a simple way some properties of Kurepa’s function, also called the left factorial. We introduce Kurepa’s hypothesis, propose a new description, and the relation to Bezout’s parameters and the Diophantine equation. A numerical analysis supports Kurepa’s hypothesis and the conjecture about distribution for Kurepa’s function.
Source:
Advances in Number Theory and Applied Analysis, 2023, 31-50
TY - CHAP
AU - Fabiano, Nicola
AU - Mirkov, Nikola S.
AU - Mitrović, Zoran D.
AU - Radenović, Stojan
PY - 2023
UR - https://vinar.vin.bg.ac.rs/handle/123456789/11359
AB - We discuss and give some insights on the Collatz conjecture, known as 3N + 1, and Kurepa’s hypothesis on the left factorial. First, the Collatz conjecture is considered and the density of values is compared to Planck’s black body radiation in physics, showing a remarkable agreement between the two. We also briefly discuss a generalization of Collatz conjecture for a generic sequence qN +1 by means of numerical analysis. Then, we give a brief historical excursus and prove in a simple way some properties of Kurepa’s function, also called the left factorial. We introduce Kurepa’s hypothesis, propose a new description, and the relation to Bezout’s parameters and the Diophantine equation. A numerical analysis supports Kurepa’s hypothesis and the conjecture about distribution for Kurepa’s function.
T2 - Advances in Number Theory and Applied Analysis
T1 - Collatz Hypothesis and Kurepa’s Conjecture
SP - 31
EP - 50
DO - 10.1142/9789811272608_0003
ER -
@inbook{
author = "Fabiano, Nicola and Mirkov, Nikola S. and Mitrović, Zoran D. and Radenović, Stojan",
year = "2023",
abstract = "We discuss and give some insights on the Collatz conjecture, known as 3N + 1, and Kurepa’s hypothesis on the left factorial. First, the Collatz conjecture is considered and the density of values is compared to Planck’s black body radiation in physics, showing a remarkable agreement between the two. We also briefly discuss a generalization of Collatz conjecture for a generic sequence qN +1 by means of numerical analysis. Then, we give a brief historical excursus and prove in a simple way some properties of Kurepa’s function, also called the left factorial. We introduce Kurepa’s hypothesis, propose a new description, and the relation to Bezout’s parameters and the Diophantine equation. A numerical analysis supports Kurepa’s hypothesis and the conjecture about distribution for Kurepa’s function.",
journal = "Advances in Number Theory and Applied Analysis",
booktitle = "Collatz Hypothesis and Kurepa’s Conjecture",
pages = "31-50",
doi = "10.1142/9789811272608_0003"
}
Fabiano, N., Mirkov, N. S., Mitrović, Z. D.,& Radenović, S.. (2023). Collatz Hypothesis and Kurepa’s Conjecture. in Advances in Number Theory and Applied Analysis, 31-50.
https://doi.org/10.1142/9789811272608_0003
Fabiano N, Mirkov NS, Mitrović ZD, Radenović S. Collatz Hypothesis and Kurepa’s Conjecture. in Advances in Number Theory and Applied Analysis. 2023;:31-50.
doi:10.1142/9789811272608_0003 .
Fabiano, Nicola, Mirkov, Nikola S., Mitrović, Zoran D., Radenović, Stojan, "Collatz Hypothesis and Kurepa’s Conjecture" in Advances in Number Theory and Applied Analysis (2023):31-50,
https://doi.org/10.1142/9789811272608_0003 . .
