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

TL;DR

This paper proves the existence of integers within certain large intervals that avoid two specific residue classes modulo primes, using analytic number theory to analyze the density and distribution of such integers.

Contribution

It introduces a novel proof demonstrating the existence of integers excluding two residue classes in specified ranges, advancing understanding of modular residue distributions.

Findings

Bounded the proportion of integers excluded by residue classes

Established the density of residue class coverage in large intervals

Proved the existence of integers avoiding two residue classes in specified ranges

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.