## A course in number theoryIdeal for undergraduate and graduate students in pure mathematics, this new text introduces the central, fundamental topics in number theory, including: divisibility and multiplicative functions; congruence and quadratic residues; continued fractions, diophantine approximation and transcendence; partitions; and diophantine equations and elliptic curves. In addition, some more advanced results are given, such as the Gelfond-Schneider theorem, the prime number theorem, and the Mordell-Weil theorem. Based on twenty years of teaching number theory, the approach is thoroughly classroom tested. Each chapter concludes with an extensive selection of sample problems, and an appendix contains hints, sketch solutions, and useful tables, making this the perfect self-contained tool for teaching and learning number theory. |

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### Contents

DIVISIBILITY | 1 |

MULTIPLICATIVE FUNCTIONS | 16 |

CONGRUENCE THEORY | 30 |

Copyright | |

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algebraic integer algebraic number field argument assume called Chapter 12 character modulo Chinese remainder theorem class group complex numbers congruence conjecture conjugate consider continued fraction representation convergent Corollary Deduce defined degree denote the number derive Diophantine equations Dirichlet's theorem distinct divisible divisors elements elliptic curve equivalence classes Euclidean algorithm Euler's example exist satisfying Fermat's finite forms with discriminant formula function Further genus given gives Hence identity implies induction inequality infinitely integer coefficients integer solutions Jacobi symbol Legendre symbol Lemma matrix method modm modp modulo Mordell Mordell-Weil theorem multiplicative non-trivial non-zero Note number of partitions number of solutions number theory odd prime p-adic Pell's equation positive integer prime factors prime number primitive root proof of Theorem properties prove quadratic form quadratic residue rational integer rational numbers real numbers result follows similar soluble square square-free Suppose Theorem 1.1 unique factorization Z/mZ zero