A note on Linnik's Theorem on quadratic non-residues

TL;DR

This paper provides a concise, combinatorial proof of Linnik's theorem, establishing bounds on the distribution of quadratic non-residues modulo primes with a focus on their size relative to prime bounds.

Contribution

It introduces a new, purely combinatorial proof of Linnik's theorem, simplifying the understanding of quadratic non-residues distribution.

Findings

Bound on the number of primes with large quadratic non-residues

Explicit constant $C_ ext{ε}$ depending on ε

Simplified proof technique

Abstract

We present a short, self-contained, and purely combinatorial proof of Linnik's theorem: for any $\varepsilon > 0$ there exists a constant $C_\varepsilon$ such that for any $N$, there are at most $C_\varepsilon$ primes $p \leqslant N$ such that the least positive quadratic non-residue modulo $p$ exceeds $N^\varepsilon$.