Density of solutions to quadratic congruences
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 prime factors for . 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 with prime factors such that a fixed quadratic equation has exactly solutions modulo .