## Lectures on Number TheoryDuring the academic year 1916-1917 I had the good fortune to be a student of the great mathematician and distinguished teacher Adolf Hurwitz, and to attend his lectures on the Theory of Functions at the Polytechnic Institute of Zurich. After his death in 1919 there fell into my hands a set of notes on the Theory of numbers, which he had delivered at the Polytechnic Institute. This set of notes I revised and gave to Mrs. Ferentinou-Nicolacopoulou with a request that she read it and make relevant observations. This she did willingly and effectively. I now take advantage of these few lines to express to her my warmest thanks. Athens, November 1984 N. Kritikos About the Authors ADOLF HURWITZ was born in 1859 at Hildesheim, Germany, where he attended the Gymnasium. He studied Mathematics at the Munich Technical University and at the University of Berlin, where he took courses from Kummer, Weierstrass and Kronecker. Taking his Ph. D. under Felix Klein in Leipzig in 1880 with a thes i s on modul ar funct ions, he became Pri vatdozent at Gcitt i ngen in 1882 and became an extraordinary Professor at the University of Konigsberg, where he became acquainted with D. Hilbert and H. Minkowski, who remained lifelong friends. He was at Konigsberg until 1892 when he accepted Frobenius' chair at the Polytechnic Institute in Z~rich (E. T. H. ) where he remained the rest of his 1 i fe. |

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### Contents

1 | |

CONGRUENCES | 51 |

LINEAR CONGRUENCES | 68 |

Decomposition of a Fraction into a Sum | 76 |

CONGRUENCES OF HIGHER DEGREE | 89 |

QUADRATIC RESIDUES | 109 |

BINARY QUADRATIC FORMS | 157 |

Reductions of the First Basic Problem of 46 | 174 |

The Primitive Representations of an odd Integer by x3 + y | 203 |

The Representation of an Integer m by a Complete System of Forms with given Discriminant A K 0 | 205 |

Regular Continued Fractions | 213 |

Equivalence of Real Irrational Numbers | 219 |

Reduced Quadratic Forms with Discriminant A 0 | 226 |

The Period of a Reduced Quadratic Form With A 0 | 232 |

Development of VAT in a Continued Fraction | 241 |

Equivalence of a form with itself and solution of the equation of Fermat for forms with Positive Discriminant A | 243 |

Reduced Quadratic Forms with Discriminant A K 0 | 184 |

The Roots of a Quadratic Form | 187 |

The Equation of Fermat and of Pell and Lagrange | 192 |

The Divisors of a Quadratic Form | 198 |

Equivalence of a form with itself and solution of the Equation of Fermat for Forms with Negative Discriminant A | 201 |

Problems for Chapter 6 | 252 |

265 | |

272 | |

### Other editions - View all

Lectures on Number Theory Adolf Hurwitz,Nikolaos Kritikos,William C Schulz No preview available - 1985 |

### Common terms and phrases

Adolf Hurwitz algorithm belongs Consequently Corollary corresponding decomposition determine Diophantine equation divisible e e e elements equal equation of Fermat example exist exponent form f forms with discriminant formula greatest common divisor Hence Hint hypothesis identically implies incongruent mod incongruent solutions irrational number least common multiple least positive lemma Let f linear congruence mathematical induction matrix mod 2k mod p-1 modulo natural number number of incongruent number of solutions odd prime pair period positive discriminant positive integer power residues mod prime number primitive representations primitive root proof is immediate proved Pythagorean triple quadratic non-residue quadratic residue mod rational number real number reduced quadratic forms reduced system regular continued fraction relatively prime residue class right adjacent satisfy Show ſº solvable square square free system of residues unimodular substitutions