## Abstract

Given a negative, we give a new upper bound on the number of square-free integers ≪X which are represented by some but not all forms of the genus of a primitive, positive-definite binary quadratic form of discriminant D. We also give an analogous upper bound for square-free integers of the form q+a≪X, where q is prime and is fixed. Combined with the-dimensional sieve of Iwaniec, this yields a lower bound on the number of such integers q+a≪X represented by a binary quadratic form of discriminant D, where D is allowed to grow with X as above. An immediate consequence of this, coming from recent work of Bourgain and Fuchs in [3], is a lower bound on the number of primes which come up as curvatures in a given primitive integer Apollonian circle packing.

Original language | English (US) |
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Pages (from-to) | 5505-5553 |

Number of pages | 49 |

Journal | International Mathematics Research Notices |

Volume | 2012 |

Issue number | 24 |

DOIs | |

State | Published - Jan 1 2012 |

Externally published | Yes |

## ASJC Scopus subject areas

- Mathematics(all)