On The Prime Numbers In Intervals

TL;DR

This paper investigates the distribution of prime numbers within specific intervals, generalizing Bertrand's postulate, exploring Legendre's conjecture, and providing explicit bounds and counts for primes in various ranges using elementary and analytic methods.

Contribution

It extends known prime distribution results to broader intervals, offers new bounds related to Legendre's conjecture, and quantifies the number of primes in these intervals with explicit lower bounds.

Findings

Proved existence of primes in intervals between kn and (k+1)n for 2 ≤ k ≤ 8 using elementary methods.

Established that for 2 ≤ k ≤ 519, there is always a prime between kn and (k+1)n for n ≥ k.

Showed a prime exists between n^2 and (n+1)^{2.000001} for all n ≥ 1.

Abstract

Bertrand's postulate establishes that for all positive integers $n>1$ there exists a prime number between $n$ and $2n$. We consider a generalization of this theorem as: for integers $n\geq k\geq 2$ is there a prime number between $kn$ and $(k+1)n$? We use elementary methods of binomial coefficients and the Chebyshev functions to establish the cases for $2\leq k\leq 8$. We then move to an analytic number theory approach to show that there is a prime number in the interval $(kn, (k+1)n)$ for at least $n\geq k$ and $2\leq k\leq 519$. We then consider Legendre's conjecture on the existence of a prime number between $n^2$ and $ (n+1)^2$ for all integers $n\geq 1$. To this end, we show that there is always a prime number between $n^2$ and $(n+1)^{2.000001}$ for all $n\geq 1$. Furthermore, we note that there exists a prime number in the interval $[n^2,(n+1)^{2+\varepsilon}]$ for any $\varepsilon>0$ and $n$ sufficiently large. We also consider the question of how many prime numbers there are between $n$ and $kn$ for positive integers $k$ and $n$ for each of our results and in the general case. Furthermore, we show that the number of prime numbers in the interval $(n,kn)$ is increasing and that there are at least $k-1$ prime numbers in $(n,kn)$ for $n\geq k\geq 2$. Finally, we compare our results to the prime number theorem and obtain explicit lower bounds for the number of prime numbers in each of our results.