Lectures on Number Theory

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Springer Science & Business Media, Dec 6, 2012 - Mathematics - 273 pages
During 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

BASIC CONCEPTS AND PROPOSITIONS
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
BIBLIOGRAPHY
265
INDEX
272
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