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

TL;DR

This paper provides an elementary proof that for sufficiently large n, there are at least n primes between the (n+1)th prime and the product of the first n+1 primes, extending previous results.

Contribution

It introduces a simple counting method combined with Stirling's formula to establish bounds on the number of primes in a specific interval, generalizing Cooke's recent result.

Findings

Proves at least n primes exist between the (n+1)th prime and the product of first n+1 primes for large n.

Establishes a general estimate for the number of primes in a specific interval based on a parameter α.

Provides an elementary proof using combinatorial and asymptotic techniques.

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.