## The New Book of Prime Number RecordsThis text originated as a lecture delivered November 20, 1984, at Queen's University, in the undergraduate colloquium senes. In another colloquium lecture, my colleague Morris Orzech, who had consulted the latest edition of the Guinness Book of Records, reminded me very gently that the most "innumerate" people of the world are of a certain trible in Mato Grosso, Brazil. They do not even have a word to express the number "two" or the concept of plurality. "Yes, Morris, I'm from Brazil, but my book will contain numbers different from ·one.''' He added that the most boring 800-page book is by two Japanese mathematicians (whom I'll not name) and consists of about 16 million decimal digits of the number Te. "I assure you, Morris, that in spite of the beauty of the appar ent randomness of the decimal digits of Te, I'll be sure that my text will include also some words." And then I proceeded putting together the magic combina tion of words and numbers, which became The Book of Prime Number Records. If you have seen it, only extreme curiosity could impel you to have this one in your hands. The New Book of Prime Number Records differs little from its predecessor in the general planning. But it contains new sections and updated records. |

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### Contents

1 | |

10 | |

Generation of Infinite Sequences of Pairwise Relatively Prime Integers | 17 |

The Power of a Prime Dividing a Factorial | 30 |

XII | 46 |

CHAPTER 3 | 180 |

Functions Satisfying Condition | 186 |

CHAPTER 4 | 213 |

CHAPTER 6 | 371 |

The Density of the Set of Regular Primes | 414 |

Conclusion | 427 |

132 | 457 |

136 | 464 |

153 | 476 |

488 | |

196 | 498 |

Interlude | 258 |

Addendum on kTuples of Primes | 265 |

The WaringGoldbach Problem | 299 |

The Distribution of Pseudoprimes Carmichael Numbers | 311 |

CHAPTER 5 | 323 |

Primes and SecondOrder Linear Recurrence Sequences | 361 |

The Pages That Couldnt Wait | 509 |

197 | 513 |

526 | |

535 | |

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### Common terms and phrases

Acta Arithm algorithm Amer arithmetic progression Assume Brillhart calculations Carmichael numbers Chapter class number Comp composite integers composite numbers congruence conjecture consecutive primes constant cyclotomic denote the number digits distinct primes divisors Dubner elliptic curves Erdös Euler exist infinitely exists a prime Fermat numbers Fermat's last theorem Fermat's little theorem finite gcd(a gcd(n hence infinitely many integers infinitely many primes integral coefficients large numbers Lehmer Lenstra log log Lucas sequences Math Mersenne numbers Mersenne primes natural number Note number of primes Number Theory odd perfect numbers odd prime paper Pomerance primality testing prime factors prime number theorem prime values primes in arithmetic primitive prime factor primitive root modulo Proc proved quadratic fields regular primes relatively prime repunits result Riemann Riemann's hypothesis Rotkiewicz satisfied Schinzel Section showed Sierpiński Sophie Germain primes sufficiently large TABLE tion twin primes Wagstaff Waring's problem Wilson primes