Geodesic
Density of solutions to quadratic congruences
Czechoslovak Mathematical Journal, Tome 67 (2017) no. 2, pp. 439-455
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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 $n\leq x$ with $k$ prime factors such that a fixed quadratic equation has exactly $2^k$ solutions modulo $n$.
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 $n\leq x$ with $k$ prime factors such that a fixed quadratic equation has exactly $2^k$ solutions modulo $n$.
DOI : 10.21136/CMJ.0.0712-15
Classification : 11B25, 11D45, 11N37
Keywords: Dirichlet's theorem; asymptotic density; primes in arithmetic progression; squarefree number 
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Prabhu, Neha. Density of solutions to quadratic congruences. Czechoslovak Mathematical Journal, Tome 67 (2017) no. 2, pp. 439-455. doi: 10.21136/CMJ.0.0712-15

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