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Proof of Existence of Integers Excluding Two Residue Values in a Specific Range

Liang Zhao

Abstract

This paper investigates the existence of integers that exclude two specific residence values modulo primes up to $p_k$ within the interval $[p_k^2, p_{k+1}^2]$. Using asymptotic results from analytic number theory, we establish bounds on the proportion of integers excluded by the union of residue classes. The findings highlight the density of residue class coverage in large intervals, contributing to the understanding of modular systems and their implications in number theory and related fields.

Proof of Existence of Integers Excluding Two Residue Values in a Specific Range

Abstract

This paper investigates the existence of integers that exclude two specific residence values modulo primes up to within the interval . Using asymptotic results from analytic number theory, we establish bounds on the proportion of integers excluded by the union of residue classes. The findings highlight the density of residue class coverage in large intervals, contributing to the understanding of modular systems and their implications in number theory and related fields.
Paper Structure (9 sections, 2 theorems, 42 equations)

This paper contains 9 sections, 2 theorems, 42 equations.

Table of Contents

  1. Introduction
  2. A Mertens--Type Formula for $\prod_{2<p<x}(1 - 2/p)$, Excluding $p=2$
  3. A Telescoping Difference for $p_{k+1}^2 P(p_{k+1}^2) \;-\; p_{k}^2 P(p_{k}^2)$, Excluding $p=2$
  4. Discussion
  5. Conclusion

Key Result

Theorem 1

Let Then there is a positive constant $\widetilde{C}_{2}$ such that, for $x\to\infty$, Moreover, where $\,M_{1}\approx0.2614972$ is the usual Meissel--Mertens constant for all primes and $S=\sum_{p}1/p^{2}\approx0.4522474$.

Theorems & Definitions (3)

  • Theorem 1
  • proof
  • Theorem 2