2017/06/04 by Kyle D. Balliet, Balliet, Kyle D.
Mathematics · #Advanced Mathematical Identities #Analytic Number Theory Research #FOS: Mathematics #History and Theory of Mathematics #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.1706.01009
openalex publication_date 2017/06/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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≥ k≥ 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≤ k≤ 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≥ k and 2≤ k≤ 519. We then consider Legendre's conjecture on the existence of a prime number between n2 and (n+1)2 for all integers n≥ 1. To this end, we show that there is always a prime number between n2 and (n+1)2.000001 for all n≥ 1. Furthermore, we note that there exists a prime number in the interval [n2,(n+1)2+ε] for any ε>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≥ k≥ 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.