On the Prime Numbers in the Interval [4n,5n]

TL;DR

This paper proves that for all sufficiently large n, there is always at least one prime between 4n and 5n, and at least seven primes between n and 5n for n > 5, addressing a specific prime distribution question.

Contribution

It establishes new bounds on the existence and quantity of primes within specific intervals scaled by n, extending prime distribution knowledge.

Findings

At least one prime exists between 4n and 5n for all n > 2.

There are at least seven primes between n and 5n for all n > 5.

The results confirm conjectures about prime distribution in scaled intervals.

Abstract

Is it true that for all $n\geq k\geq 2$ there exists a prime number between $kn$ and $(k+1)n$? In this paper we show that there is always a prime number between $4n$ and $5n$ for all $n>2$. We also show there are at least seven prime numbers between $n$ and $5n$ for all $n>5$.