On the Zeckendorf representation of smooth numbers

TL;DR

This paper investigates the properties of Zeckendorf representations of integers, showing that large integers cannot be both divisible only by small primes and have a sparse Zeckendorf representation.

Contribution

It provides a quantitative analysis linking prime divisibility constraints with the sparsity of Zeckendorf representations in large integers.

Findings

Large integers cannot be divisible only by small primes and have few Zeckendorf digits

Quantitative bounds on the Zeckendorf representation for integers with prime divisibility constraints

Establishment of limitations on the structure of integers based on their prime factors and Zeckendorf form

Abstract

Among other results, we establish, in a quantitative form, that any sufficiently large integer cannot simultaneously be divisible only by very small primes and have very few digits in its Zeckendorf representation.