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arXiv · 1211.4571

An elemetary proof of an estimate for a number of primes less than the product of the first $n$ primes

Romeo Meštrović
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Abstract

Let $\alpha$ be a real number such that $1< \alpha <2$ and let $x_0=x_0(\alpha)$ be a {\rm(}unique{\rm)} positive solution of the equation $$ x^{\alpha-1} -\frac{\pi}{e^2\sqrt{3}}x +1=0. $$ Then we prove that for each positive integer $n>x_0$ there exist at least $[n^\alpha]$ primes between the $(n+1)$th prime and the product of the first $n+1$ primes. In particular, we establish a recent Cooke's result which asserts that for each positive integer $n$ there are at least $n$ primes between the $(n+1)$th prime and the product of the first $n+1$ primes. Our proof is based on an elementary counting method (enumerative arguments) and the application of Stirling's formula to give upper bound for some binomial coefficients.

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Romeo Meštrović. 2012-11-17. An elemetary proof of an estimate for a number of primes less than the product of the first $n$ primes. https://arxiv.org/abs/1211.4571

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