On the Primes in the Interval [3n, 4n]
Number Theory
2011-10-12 v1
Abstract
For the old question whether there is always a prime in the interval [kn, (k+1)n] or not, the famous Bertrand's postulate gave an affirmative answer for k=1. It was first proved by P.L. Chebyshev in 1850, and an elegant elementary proof was given by P. Erdos in 1932. M. El Bachraoui used elementary techniques to prove the case k=2 in 2006. This paper gives a proof of the case k=3, again without using the prime number theorem or any deep analytic result. In addition we give a lower bound for the number of primes in the interval [3n, 4n], which shows that as n tends to infinity, the number of primes in the interval [3n, 4n] goes to infinity.
Cite
@article{arxiv.1110.2377,
title = {On the Primes in the Interval [3n, 4n]},
author = {Andy Loo},
journal= {arXiv preprint arXiv:1110.2377},
year = {2011}
}