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

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.