## A Course in Number TheoryThis textbook covers the main topics in number theory as taught in universities throughout the world. Number theory deals mainly with properties of integers and rational numbers; it is not an organized theory in the usual sense but a vast collection of individual topics and results, with some coherent sub-theories and a long list of unsolved problems. This book excludes topics relying heavily on complex analysis and advanced algebraic number theory. The increased use of computers in number theory is reflected in many sections (with much greater emphasis in this edition). Some results of a more advanced nature are also given, including the Gelfond-Schneider theorem, the prime number theorem, and the Mordell-Weil theorem. The latest work on Fermat's last theorem is also briefly discussed. Each chapter ends with a collection of problems; hints or sketch solutions are given at the end of the book, together with various useful tables. |

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### Contents

DiVisiBility | 1 |

MULTIPLICATIVE FUNCTIONS | 17 |

CONGRUENCE THEORY | 32 |

QUADRATIC RESIDUES | 58 |

ALGEBRAIC TOPICS | 80 |

SUMS OF SQUARES AND GAUSS SUMS | 102 |

CONTINUED FRACTIONS | 124 |

TRANSCENDENTAL NUMBERS | 146 |

PARTitions | 211 |

the PRiME NUMBERS | 226 |

TWO MAJOR THEOREMS ON THE PRIMES | 247 |

DIOPHANTINE EQUATIONS | 273 |

BASIC THEORY | 293 |

FURTHER RESULTS | 317 |

ANSWERS AND HINTS TO PROBLEMS | 347 |

BiBLIOGRAPHY | 389 |

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algebraic integer algebraic number algebraic number field algorithm assume Chapter 12 character modulo Chinese remainder theorem congruence conjecture conjugate consider continued fraction continued fraction representation convergent coordinates Corollary Deduce defined definition derive Diophantine equations divisors elements elliptic curve Euler's example exist satisfying Fermat's form f forms with discriminant formula function Further Gauss given gives Hence identity implies induction inequality infinitely integer solutions Jacobi symbol Legendre symbol Lemma Let f matrix method modulo Mordell Mordell–Weil Mordell–Weil theorem multiplicative nonzero Note number field number of partitions number of solutions number theory odd prime p-adic Pell's equation positive integer prime factors prime number primitive root Problem proof of Theorem properties prove quadratic form quadratic residues rational integer rational numbers real numbers result follows Secondly Section similar soluble square square-free Suppose Theorem 1.1 torsion Z/pZ zero