The Distribution of Self-Fibonacci Divisors

TL;DR

This paper investigates the distribution of positive integers that divide their corresponding Fibonacci number, establishing an upper bound on their counting function that involves iterated logarithms.

Contribution

It provides a new upper bound on the count of integers dividing their Fibonacci number, advancing understanding of Fibonacci divisibility patterns.

Findings

Established an upper bound involving iterated logarithms

Improved understanding of Fibonacci divisor distribution

Quantified the growth rate of the counting function

Abstract

Consider the positive integers $n$ such that $n$ divides the $n$-th Fibonacci number, and their counting function $A$. We prove that \[A(x) \leq x^{1-(1/2+o(1))\log\log\log x/\log\log x}.\]