A Note on Prime Fibonacci Sequences

TL;DR

This paper introduces prime Fibonacci sequences, a new Fibonacci-like sequence variant where each term is the smallest odd prime divisor of the sum of the previous two, and proves their termination in powers of two.

Contribution

It defines prime Fibonacci sequences and proves their termination properties, extending the understanding of Fibonacci-like sequences with prime divisor conditions.

Findings

Sequences always terminate in a power of two

Sequences can be extended infinitely to the left

Provides mathematical proof of termination behavior

Abstract

In this paper, we define a variant of Fibonacci-like sequences that we call prime Fibonacci sequences, where one takes the sum of the previous two terms and returns the smallest odd prime divisor of that sum as the next term. We prove that these sequences always terminate in a power of two but can be extended infinitely to the left.