Another inequality inspired by Erdős

TL;DR

The paper studies two Erdős-inspired inequalities aiming for arithmetically pure proofs of a Bertrand postulate variant. It defines arithmetic parameters $z(n)$, $m(n)$, $r(n)$ and associated sequences, and analyzes the sign of $y(n)=2^{2n-2z(n)-m(n)+2}-n^{m(n)-1}$ alongside the earlier quantity $x(n)=z(n)-(r(n)+1)m(n)$. Section 2 delivers a purely arithmetic, interval-based solution that yields a complete sign distribution for $y(n)$ (Theorem T2) and confirms $y(n)\neq 0$ for all $n$, while Section 3 provides a calculus-based approach with a convex auxiliary function to bound where the inequalities hold, obtaining $n\le 560$ for (1) and $n\ge 561$ implying $x(n)>0$, together with explicit zeroes of $x(n)$ at $n\in\{436,451,529,545,546\}$. The combination of arithmetic partitioning and elementary calculus demonstrates both inequalities with elementary methods and identifies critical zero cases. The results contribute a pedagogical bridge between Erdős-style proofs and arithmetical techniques for postulate-like statements.

Abstract

In our effort to find an arithmetically pure proof of the Bertrand postulate, we investigate and solve (using only elementary arithmetical methods) another less usual inequality in positive integers inspired by the classical proof of the postulate given by P. Erdős.

Theorems & Definitions (27)

• Theorem 1.1
• Lemma 1.3
• proof
• Lemma 2.3
• proof
• Lemma 2.4
• proof
• Lemma 2.5
• proof
• Proposition 2.6