# Density of solutions to quadratic congruences

• Volume: 67, Issue: 2, page 439-455
• ISSN: 0011-4642

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## Abstract

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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 $n\le x$ with $k$ prime factors such that a fixed quadratic equation has exactly ${2}^{k}$ solutions modulo $n$.

## How to cite

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Prabhu, Neha. "Density of solutions to quadratic congruences." Czechoslovak Mathematical Journal 67.2 (2017): 439-455. <http://eudml.org/doc/288207>.

@article{Prabhu2017,
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 $n\le x$ with $k$ prime factors such that a fixed quadratic equation has exactly $2^k$ solutions modulo $n$.},
author = {Prabhu, Neha},
journal = {Czechoslovak Mathematical Journal},
keywords = {Dirichlet's theorem; asymptotic density; primes in arithmetic progression; squarefree number},
language = {eng},
number = {2},
pages = {439-455},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Density of solutions to quadratic congruences},
url = {http://eudml.org/doc/288207},
volume = {67},
year = {2017},
}

TY - JOUR
AU - Prabhu, Neha
TI - Density of solutions to quadratic congruences
JO - Czechoslovak Mathematical Journal
PY - 2017
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 67
IS - 2
SP - 439
EP - 455
AB - 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 $n\le x$ with $k$ prime factors such that a fixed quadratic equation has exactly $2^k$ solutions modulo $n$.
LA - eng
KW - Dirichlet's theorem; asymptotic density; primes in arithmetic progression; squarefree number
UR - http://eudml.org/doc/288207
ER -

## References

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1. Hardy, G. H., Wright, E. M., An Introduction to the Theory of Numbers, Oxford University Press, Oxford (2008). (2008) Zbl1159.11001MR2445243
2. Kornblum, H., Landau, E., 10.1007/BF01203156, Math. Zeitschr. 5 (1919), 100-111 German. (1919) Zbl47.0154.02MR1544375DOI10.1007/BF01203156
3. Landau, E., Sur quelques problèmes relatifs à la distribution des nombres premiers, S. M. F. Bull. 28 (1900), 25-38 French. (1900) Zbl31.0200.01MR1504359
4. Montgomery, H. L., Vaughan, R. C., 10.1017/CBO9780511618314, Cambridge Studies in Advanced Mathematics 97, Cambridge University Press, Cambridge (2007). (2007) Zbl1142.11001MR2378655DOI10.1017/CBO9780511618314
5. Pomerance, C., 10.1515/crll.1977.293-294.217, J. Reine Angew. Math. 293/294 (1977), 217-222. (1977) Zbl0349.10004MR0447087DOI10.1515/crll.1977.293-294.217
6. Ribenboim, P., 10.1007/978-1-4612-0759-7, Springer, New York (1996). (1996) Zbl0856.11001MR1377060DOI10.1007/978-1-4612-0759-7
7. Wright, E. M., 10.1017/S0013091500021349, Proc. Edinb. Math. Soc., II. Ser. 9 (1954), 87-90. (1954) Zbl0057.28601MR0065579DOI10.1017/S0013091500021349

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