Density of solutions to quadratic congruences

TL;DR

This paper extends classical number theory results by determining the density of composite numbers with a fixed number of prime factors satisfying certain congruences, and applies this to quadratic equations modulo these numbers.

Contribution

It generalizes Dirichlet's theorem to numbers with multiple prime factors and computes their densities under specific conditions, building on Wright's 1954 proof.

Findings

Density of such numbers is explicitly computed.

Application to quadratic equations shows exact number of solutions.

Provides a framework for understanding distributions of composite numbers with prime factor constraints.

Abstract

A classical result in number theory is Dirichlet's theorem on the density of primes in an arithmetic progression. We prove a similar result for numbers with exactly k prime factors for k>1. Building upon a proof by E.M. Wright in 1954, we compute the natural density of such numbers where each prime satisfies a congruence condition. As an application, we obtain the density of squarefree n not greater than x with k prime factors such that a fixed quadratic equation has exactly 2^k solutions modulo n.