## Advisor(s)

Mihai Caragiu, PhD

Ohio Northern University

Mathematics, Science, Technology, and Mathematics

m-caragiu.1@onu.edu

## Document Type

Poster

## Location

ONU McIntosh Center; Activities Room

## Start Date

22-4-2022 11:00 AM

## End Date

22-4-2022 12:00 PM

## Abstract

The greatest prime factor sequences (GPF sequences), born at ONU in 2005, are integer sequences satisfying recursions in which every term is the greatest prime factor of a linear combination with positive integer coefficients of the preceding *k* terms (where *k* is the order of the sequence), possibly including a positive constant. The very first GPF sequence that was introduced satisfies the recursion x(n+1)=GPF(2*x(n)+1). In 2005 it was conjectured that no matter the seed, this particular GPF sequence enters the limit cycle (attractor) 3-7-5-11-23-47-19-13. In our current work, of a computational nature, we introduce the functions “depth” – where depth(n) is the number of iterations needed to reach the attractor given that the seed is x(0)=n, and “land” – where land(n) is the specific element of the attractor where the entrance is made, again, given that the seed is x(o)=n. We investigate the functions depth(n) and land(n), and calculate specific values using the MAPLE computer algebra system. This turns out to be a very interesting example of “arithmetic dynamics”.

## Recommended Citation

Hare, Alexander, "A strange attractor of primes" (2022). *ONU Student Research Colloquium*. 52.

https://digitalcommons.onu.edu/student_research_colloquium/2022/posters/52

A strange attractor of primes

ONU McIntosh Center; Activities Room

The greatest prime factor sequences (GPF sequences), born at ONU in 2005, are integer sequences satisfying recursions in which every term is the greatest prime factor of a linear combination with positive integer coefficients of the preceding *k* terms (where *k* is the order of the sequence), possibly including a positive constant. The very first GPF sequence that was introduced satisfies the recursion x(n+1)=GPF(2*x(n)+1). In 2005 it was conjectured that no matter the seed, this particular GPF sequence enters the limit cycle (attractor) 3-7-5-11-23-47-19-13. In our current work, of a computational nature, we introduce the functions “depth” – where depth(n) is the number of iterations needed to reach the attractor given that the seed is x(0)=n, and “land” – where land(n) is the specific element of the attractor where the entrance is made, again, given that the seed is x(o)=n. We investigate the functions depth(n) and land(n), and calculate specific values using the MAPLE computer algebra system. This turns out to be a very interesting example of “arithmetic dynamics”.