### Abstract

We begin by reviewing various classical problems concerning the existence of primes or numbers with few prime factors as well as some of the key developments towards resolving these long standing questions. Then we put the theory in a natural and general geometric context of actions on affine n-space and indicate what can be established there. The methods used to develop a combinational sieve in this context involve automorphic forms, expander graphs and unexpectedly arithmetic combinatorics. Applications to classical problems such as the divisibility of the areas of Pythagorean triangles and of the curvatures of the circles in an integral Apollonian packing, are given.

Original language | English (US) |
---|---|

Title of host publication | Differential Geometry, Mathematical Physics, Mathematics and Society Part 2 |

Pages | 225-240 |

Number of pages | 16 |

Edition | 322 |

State | Published - Dec 1 2008 |

### Publication series

Name | Asterisque |
---|---|

Number | 322 |

ISSN (Print) | 0303-1179 |

### All Science Journal Classification (ASJC) codes

- Mathematics(all)

### Keywords

- Affine orbits
- Expanders and sumproduct
- Primes
- Saturation numbers
- Sieves

## Fingerprint Dive into the research topics of 'Equidistribution and primes'. Together they form a unique fingerprint.

## Cite this

*Differential Geometry, Mathematical Physics, Mathematics and Society Part 2*(322 ed., pp. 225-240). (Asterisque; No. 322).