## A Pathway Into Number TheoryNumber theory is concerned with the properties of the natural numbers: 1,2,3,.... During the seventeenth and eighteenth centuries, number theory became established through the work of Fermat, Euler and Gauss. With the hand calculators and computers of today, the results of extensive numerical work are instantly available and mathematicians may traverse the road leading to their discoveries with comparative ease. Now in its second edition, this book consists of a sequence of exercises that will lead readers from quite simple number work to the point where they can prove algebraically the classical results of elementary number theory for themselves. A modern high school course in mathematics is sufficient background for the whole book which, as a whole, is designed to be used as an undergraduate course in number theory to be pursued by independent study without supporting lectures. |

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With problems ordered in a discovery fashion and historical background behind each topic, this book presents a great way to teach number theory. The book contains a fair amount of interesting questions as well; however, it lacks a big and necessary portion of a math textbook: the information and worked examples. Truly a "pathway" into number theory, it would work well as a study guide and exercise book, but not a textbook.

### Contents

III | 1 |

IV | 7 |

V | 9 |

VI | 13 |

VIII | 14 |

X | 16 |

XI | 22 |

XII | 27 |

XLVII | 129 |

XLVIII | 130 |

XLIX | 140 |

L | 141 |

LI | 145 |

LII | 147 |

LIV | 148 |

LV | 154 |

XIII | 33 |

XIV | 36 |

XV | 37 |

XVI | 38 |

XVII | 39 |

XVIII | 48 |

XIX | 53 |

XXI | 54 |

XXII | 55 |

XXIII | 56 |

XXIV | 61 |

XXV | 64 |

XXVI | 66 |

XXVII | 67 |

XXVIII | 68 |

XXIX | 70 |

XXX | 79 |

XXXI | 81 |

XXXII | 84 |

XXXIII | 87 |

XXXIV | 88 |

XXXV | 89 |

XXXVI | 97 |

XXXVII | 100 |

XXXVIII | 101 |

XXXIX | 102 |

XL | 108 |

XLII | 110 |

XLIII | 119 |

XLIV | 123 |

XLV | 126 |

XLVI | 127 |

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### Common terms and phrases

argument of q Chinese remainder theorem column congruent contains continued fraction convergents coprime cosets cubes Deduce definite quadratic form determine equivalent quadratic forms Euler exist integers expressed factorisation Farey sequence find integers form with discriminant four squares fundamental parallelepiped fundamental parallelogram Gaussian integers graph integral coefficients irrational number Lagrange's theorem lattice points maps matrix Minkowski's theorem mod 9 mod3 modp modulo 11 Multiplication modulo non-zero integers number of lattice number of partitions number theory numerically least residues odd number odd prime pairs parallelepiped periodic continued fraction polynomial positive integers prime factors prime number primitive root modulo prove Pythagorean triple quadratic equation quadratic irrational quadratic non-residue quadratic residue modulo rational number real number reduced form region residue classes set of residues subgroup of Z2 sum of four Theorem q triangle unimodular matrix unimodular transformation unique values zero