Schreier Numbers and Nontrivial Small Divisors Satisfying Linear Recurrence of Order at Most Two

TL;DR

This paper explores Schreier numbers, their density, and characterizes natural numbers with small divisors satisfying linear recurrences of order at most two, revealing new structural insights and infinite pairs with specific differences.

Contribution

It proves the asymptotic density of Schreier numbers is zero and characterizes numbers with small divisors following linear recurrences of order up to two.

Findings

Asymptotic density of Schreier numbers is zero

Infinitely many non-prime Schreier pairs with difference 2 or 4

Complete characterization of numbers with small divisors satisfying linear recurrence of order ≤ 2

Abstract

Schreier sets have been an object of study since first introduced in 1930 by Jozef Schreier to construct a counterexample to a conjecture of Banach. In 1974 George Andrews found interesting connections between these sets and Fibonacci number, and since then more results of a combinatorial flavor were proven by Chu, Beanland, and Finch-Smith. In parallel Iannucci introduced the concept of a small divisor and characterized all natural numbers whose small divisors are in arithmetic progression, results which were generalized by Chentouf and Chu. Then combining these two ideas, Chu introduced the notion of a Schreier number, one whose nontrivial small divisor set (small divisors excluding 1) is Schreier. Our main results are twofold: we first prove the asymptotic density of these numbers is 0 and that there are infinitely many non-prime Schreier pairs with difference 2 or 4. Then motivated by Chu's generalization of Iannucci's result we characterize all natural numbers whose nontrivial small divisors satisfy a linear recurrence with order no larger than 2.