A lower bound on the number of rough numbers

TL;DR

This paper establishes a new lower bound on the count of rough numbers, which are integers with no small prime factors, providing insights into their distribution for certain parameters.

Contribution

The paper proves a specific lower bound for the number of rough numbers, advancing understanding of their minimal quantity under given conditions.

Findings

Proves that (n,p) q; loor(2n/p) + 1 for p q; 11 and n q; 2p

Establishes bounds for rough numbers with prime thresholds p q; 11

Provides a foundational result for further research on rough number distribution

Abstract

Conceptually, a rough number is a positive integer with no small prime factors. Formally, for real numbers $x$ and $y$, let $\Phi(x,y)$ denote the number of positive integers at most $x$ with no prime factors less than $y$. In this paper we establish the lower bound $\Phi(n,p)\geq \lfloor 2n/p \rfloor +1$ when $p\geq 11$ is prime and $n\geq 2p$.