On the Primes in the Interval [3n, 4n]

Details On the Primes in the Interval [3n, 4n] On the Primes in the Interval [3n, 4n]

2011-10-11

arxiv.org/abs/1110.2377v1

International Journal of Contemporary Mathematical Sciences, Vol. 6, 2011, no. 38, 1871-1882

Mathematics

Number Theory

Scientific paper

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.

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