# Primes in arithmetic progressions and nonprimitive roots

## TL;DR

This paper investigates the distribution of primes in arithmetic progressions related to near primitive roots, extending known results about primitive roots and their subgroup indices in modular arithmetic.

## Contribution

It introduces the concept of t-near primitive roots and proves a new, more complex result about their distribution among primes.

## Key findings

- Each primitive residue class contains a positive density subset of primes without a given t-near primitive root.
- Established a density result for primes in arithmetic progressions related to t-near primitive roots.
- Extended classical results on primitive roots to the broader context of t-near primitive roots.

## Abstract

Let $p$ be a prime. If an integer $g$ generates a subgroup of index $t$ in $(\mathbb Z/p\mathbb Z)^*,$ then we say that $g$ is a $t$-near primitive root modulo $p$. We point out the easy result that each primitive residue class contains a positive natural density subset of primes $p$ not having $g$ as a $t$-near primitive root and prove a more difficult variant.