## 17 Lectures on Fermat Numbers: From Number Theory to GeometryThe pioneering work of French mathematician Pierre de Fermat has attracted the attention of mathematicians for over 350 years. This book was written in honor of the 400th anniversary of his birth, providing readers with an overview of the many properties of Fermat numbers and demonstrating their applications in areas such as number theory, probability theory, geometry, and signal processing. This book introduces a general mathematical audience to basic mathematical ideas and algebraic methods connected with the Fermat numbers. |

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### Contents

Introduction | 1 |

Fundamentals of Number Theory | 9 |

Basic Properties of Fermat Numbers | 27 |

The Most Beautiful Theorems on Fermat Numbers | 34 |

Primality of Fermat Numbers | 41 |

Divisibility of Fermat Numbers | 59 |

Factors of Fermat Numbers | 71 |

Connection of Fermat Numbers with Pascals Triangle | 81 |

Fermats Little Theorem Pseudoprimes and Superpseudoprimes | 130 |

Generalizations of Fermat Numbers | 149 |

Open Problems | 158 |

Fermat Number Transform and Other Applications | 167 |

The Proof of Gausss Theorem | 189 |

Euclidean Construction of the Regular Heptadecagon | 195 |

Appendix | 209 |

B Mersenne Numbers | 213 |

Miscellaneous Results | 94 |

The Irrationality of the Sum of Some Reciprocals | 104 |

Fermat Primes and a Diophantine Equation | 119 |

Remembrance of Pierre de Fermat | 218 |

226 | |

### Other editions - View all

17 Lectures on Fermat Numbers: From Number Theory to Geometry Michal Krizek,Florian Luca,Lawrence Somer No preview available - 2011 |

### Common terms and phrases

algebraic algorithm arithmetic assume base binary Brent Brillhart composite Fermat number composite numbers congruence coprime cyclotomic polynomials defined denote digits distinct prime divides divisible equal equation Euclidean construction Euler pseudoprime Euler's criterion exist infinitely factor of F Fermat numbers Fermat primes Fermat's last theorem Fermat's little theorem Figure finite following theorem form k2 formula Gauss given Goldbach's Theorem 4.1 Hence higher powers implies infinitely many Fermat Legendre symbol Lemma Lenstra log2 Luca Lucas's Theorem Mersenne number Mersenne primes mod F Moreover multiple natural number number F number theory odd integer odd number odd prime Pepin's test periodic solutions Pierre de Fermat positive integer primality test prime factors prime number primitive prime divisor primitive root modulo proof of Theorem Proposition proved quadratic nonresidue regular polygons Remark residue modulo result Ribenboim Rotkiewicz number ruler and compass satisfies Selfridge sequence square strong pseudoprime superpseudoprime Suppose