## High Primes and Misdemeanours: Lectures in Honour of the 60th Birthday of Hugh Cowie WilliamsHugh C. Williams, A. J. Van Der Poorten, Andreas Stein This volume consists of a selection of papers based on presentations made at the international conference on number theory held in honor of Hugh Williams' sixtieth birthday. The papers address topics in the areas of computational and explicit number theory and its applications. The material is suitable for graduate students and researchers interested in number theory. |

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### Contents

1 | |

On a Generalization of the Radix Representationa Survey | 19 |

Multiplicative Structure of Values of the Euler Function | 29 |

Congruences and Exponential Sums with the Euler Function | 49 |

On the Existence of Cyclic Difference Sets with Small Parameters | 61 |

Doubly Focused Enumeration of Locally Square Polynomial Values | 69 |

Cubic Reciprocity and Explicit Primality Tests for h3k+1 | 77 |

Algorithms for Finding Almost Irreducible and Almost | 91 |

On the Research Contributions of Hugh C Williams | 197 |

Hilbert Modular Fourfolds of Arithmetic Genus One | 217 |

Divisibility of Exponents of Class Groups of Pure Cubic Number Fields | 237 |

Hyperelliptic Curves and Cryptography | 255 |

Remarks on S Chowlas Hypothesis Implying that Ls x 0 | 283 |

Jeepers Creepers | 305 |

Algebraic Tori in Cryptography | 317 |

An Elliptic Curve Trapdoor System Extended Abstract | 341 |

Commentary on Lucas Test | 103 |

Number Field Cryptography | 111 |

Heuristics for Class Numbers of PrimePower Real Cyclotomic Fields | 149 |

On a Conjecture of Feit and Thompson | 169 |

The Cunningham Project | 367 |

Advice to Referees of Submissions | 391 |

### Common terms and phrases

algebraic algorithm amicable pairs applications arithmetic assume bits bound called class group Comp complex computation conjecture consider constant construct continued cryptography cubic defined denote determine difference discrete logarithm discriminant dividing divisible divisor efficient element elliptic curve equation example exists expected extension fact factor field fixed fraction function genus give given holds hyperelliptic curves ideal implementation integer irreducible isomorphism known Lemma log log Math Mathematics method modulo multiplication Note number fields Number Theory obtain operations performance period polynomial possible present primality prime primitive problem proof prove quadratic rational reduced relation Remark requires root running satisfying sequence sieve solutions solve square step Table takes Theorem trinomials unit values write yields

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