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One inequality inspired by Erdős

Barbora Batíková, Tomáš J. Kepka, Petr C. Němec

Abstract

Inspired by the proof of the Bertrand postulate given by P. ErdőS, we carefully examine and solve one less usual inequality in positive integers which could help to find an arithmetically pure proof that for every positive integer $n\ge2$ there is a prime $p$ such that $n<p<2n$.

One inequality inspired by Erdős

Abstract

Inspired by the proof of the Bertrand postulate given by P. ErdőS, we carefully examine and solve one less usual inequality in positive integers which could help to find an arithmetically pure proof that for every positive integer there is a prime such that .

Paper Structure

This paper contains 3 theorems, 1 equation.

Key Result

Lemma 1

Let $a,b,n$ be positive integers such that $a\le n\le b$. (i) $z(a)-(r(b)+1)m(b)\le x(n)\le z(b)-(r(a)+1)m(a)$. (ii) If $x(a)<0$ and $b$ is the largest integer such that $2b\le3(r(a)+1)m(a)$ then $x(n)<0$. (iii) If $x(b)>0$ and $a$ is the least integer with $2a\ge3(r(b)+1)m(b)+4$ then $x(n)>0$.

Theorems & Definitions (8)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Remark 1
  • Theorem 1
  • proof
  • Remark 2