### Abstract

For any nonzero h ∈ ℤ, we prove that a positive proportion of integral binary cubic forms F do locally everywhere represent h but do not globally represent h; that is, a positive proportion of cubic Thue equations F(x,y) = h fail the integral Hasse principle. Here, we order all classes of such integral binary cubic forms F by their absolute discriminants. We prove the same result for Thue equations G(x,y) = h of any fixed degree n ≥ 3, provided that these integral binary n-ic forms G are ordered by the maximum of the absolute values of their coefficients.

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

Pages (from-to) | 283-307 |

Number of pages | 25 |

Journal | American Journal of Mathematics |

Volume | 141 |

Issue number | 2 |

DOIs | |

State | Published - Apr 2019 |

### All Science Journal Classification (ASJC) codes

- Mathematics(all)

## Fingerprint Dive into the research topics of 'A positive proportion of thue equations fail the integral hasse principle'. Together they form a unique fingerprint.

## Cite this

Akhtari, S., & Bhargava, M. (2019). A positive proportion of thue equations fail the integral hasse principle.

*American Journal of Mathematics*,*141*(2), 283-307. https://doi.org/10.1353/ajm.2019.0006