Three’s A Crowd: An Exploration of Subprime Tribonacci Sequences

Abstract

AbstractA subprime Fibonacci sequence follows the Fibonacci recurrence, where the next term in a sequence is the sum of the two previous terms, except that composite sums are divided by their least prime factor. We extend the recurrence to three terms, investigating subprime tribonacci sequences. It appears that all such sequences eventually enter a repeating cycle. We compute cycles arising from more than one billion sequences, classifying them as trivial, tame, and wild. We further investigate questions of parity and primality in subprime tribonacci sequences. In particular, we show that any nonzero subprime tribonacci sequence eventually contains an odd term. AcknowledgmentThis article grew out of a final project by the first three authors in the course Modern Computational Mathematics at St. Olaf College in spring 2020.Additional informationNotes on contributorsSara BarrowsSara Barrows (sarabarrows18@gmail.com) received a BA in Physics and a concentration in Engineering Studies from St. Olaf College in 2022. She currently works in R&D for Saint-Gobain, a materials company. As a Research Engineer, Sara investigates innovative solutions within the building materials industry.Emily NoyeEmily Noye (emily.noye@yahoo.com) has her BA in Mathematics with a concentration in Statistics and Data Science from St. Olaf College. She now works as a Retirement Actuary for Aon Consulting while working toward her ASA certification.Sarah UttormarkSarah Uttormark (smu32@cornell.edu) earned a BA in Physics, Mathematics, Norwegian, and Nordic Studies with a concentration in Engineering Studies from St. Olaf College in 2022. She is now an Applied Physics graduate student at Cornell University, where she works under Professor Lois Pollack on instrumentation and methods for studying the structures and dynamics of biomolecules.Matthew WrightMatthew Wright (wright5@stolaf.edu) is an Associate Professor at St. Olaf College (Northfield, MN), where he teaches applied and computational math courses. He is an author of the RIVET software for topological data analysis. Matthew lives in Minnesota with his wife and two children, and also enjoys juggling. Find him online at mlwright.org.