## Elementary Theory of Numbers: Second English Edition (edited by A. Schinzel)Since the publication of the first edition of this work, considerable progress has been made in many of the questions examined. This edition has been updated and enlarged, and the bibliography has been revised. The variety of topics covered here includes divisibility, diophantine equations, prime numbers (especially Mersenne and Fermat primes), the basic arithmetic functions, congruences, the quadratic reciprocity law, expansion of real numbers into decimal fractions, decomposition of integers into sums of powers, some other problems of the additive theory of numbers and the theory of Gaussian integers. |

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### Contents

1 | |

32 | |

Chapter III Prime Numbers | 113 |

Chapter IV Number of Divisors and Their Sum | 166 |

Chapter V Congruences | 198 |

Chapter VI EulerS Totient Function and The Theorem of Euler | 245 |

Chapter VII Representation of Numbers Bydecimals in a Given Scale | 285 |

Chapter VIII Continued Fractions | 304 |

Chapter X Mersenne Numbers and Fermat Numbers | 360 |

Chapter XI Representations of Natural Numbers as Sums of Non negative kth Powers | 378 |

Chapter XII Some Problems of the Additive Theory of Numbers | 431 |

Chapter XIII Complex Integers | 449 |

Bibliography | 482 |

505 | |

511 | |

Addendum and Corrigendum Inserted in July 1987 | 514 |

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### Common terms and phrases

arbitrary assumption belongs calculate Chapter clearly complex integers composite condition congruence conjecture Consequently consider consisting contains contrary convergent Corollary cubes defined definition denote digits divided divisible easily easy to prove equal example Exercise exist infinitely exist natural exponent fact follows form 4k formula four function given gives greater hand Hence holds identity immediately implies impossible inequality infer least lemma less Math modulus Moreover multiple natural number necessary number n obtain odd number pairs period points positive precisely prime divisor prime factors prime numbers primitive proof proved rational numbers relation relatively prime remainder representation respect root satisfy sequence shows side solution of equation solutions solutions in natural squares sufficient summands Suppose Theorem triangles true values verify virtue whence zero