Number Theory

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Courier Corporation, Oct 12, 1994 - Mathematics - 259 pages
4 Reviews

Although mathematics majors are usually conversant with number theory by the time they have completed a course in abstract algebra, other undergraduates, especially those in education and the liberal arts, often need a more basic introduction to the topic.
In this book the author solves the problem of maintaining the interest of students at both levels by offering a combinatorial approach to elementary number theory. In studying number theory from such a perspective, mathematics majors are spared repetition and provided with new insights, while other students benefit from the consequent simplicity of the proofs for many theorems.
Among the topics covered in this accessible, carefully designed introduction are multiplicativity-divisibility, including the fundamental theorem of arithmetic, combinatorial and computational number theory, congruences, arithmetic functions, primitive roots and prime numbers. Later chapters offer lucid treatments of quadratic congruences, additivity (including partition theory) and geometric number theory.
Of particular importance in this text is the author's emphasis on the value of numerical examples in number theory and the role of computers in obtaining such examples. Exercises provide opportunities for constructing numerical tables with or without a computer. Students can then derive conjectures from such numerical tables, after which relevant theorems will seem natural and well-motivated..

 

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overall good book, should offer more background and examples similar to the homework questions asked. I'm finding this book to be interestingly challenging

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accessible for most people with a basic math vocabulary. overall interesting - especially if you like math and if you're looking for a good reference on number theory.

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Contents

MULTIPLICATIVITYDIVISIBILITY
1
BASIS REPRESENTATION
3
THE FUNDAMENTAL THEOREM OF ARITHMETIC
12
COMBINATORIAL AND COMPUTATIONAL NUMBER THEORY
30
FUNDAMENTALS OF CONGRUENCES
49
SOLVING CONGRUENCES
58
ARITHMETIC FUNCTIONS
75
PRIMITIVE ROOTS
93
QUADRATIC RESIDUES
115
DISTRIBUTION OF QUADRATIC RESIDUES
128
ADDITIVITY
139
SUMS OF SQUARES
141
ELEMENTARY PARTITION THEORY
149
PARTITION GENERATING FUNCTIONS
160
PARTITION IDENTITIES
175
GEOMETRIC NUMBER THEORY
199

PRIME NUMBERS
100
QUADRATIC CONGRUENCES
113
LATTICE POINTS
201
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About the author (1994)

The Holy Grail of Number Theory
George E. Andrews, Evan Pugh Professor of Mathematics at Pennsylvania State University, author of the well-established text Number Theory (first published by Saunders in 1971 and reprinted by Dover in 1994), has led an active career discovering fascinating phenomena in his chosen field — number theory. Perhaps his greatest discovery, however, was not solely one in the intellectual realm but in the physical world as well.

In 1975, on a visit to Trinity College in Cambridge to study the papers of the late mathematician George N. Watson, Andrews found what turned out to be one of the actual Holy Grails of number theory, the document that became known as the "Lost Notebook" of the great Indian mathematician Srinivasa Ramanujan. It happened that the previously unknown notebook thus discovered included an immense amount of Ramanujan's original work bearing on one of Andrews' main mathematical preoccupations — mock theta functions. Collaborating with colleague Bruce C. Berndt of the University of Illinois at Urbana-Champaign, Andrews has since published the first two of a planned three-volume sequence based on Ramanujan's Lost Notebook, and will see the project completed with the appearance of the third volume in the next few years.

In the Author's Own Words:
"It seems to me that there's this grand mathematical world out there, and I am wandering through it and discovering fascinating phenomena that often totally surprise me. I do not think of mathematics as invented but rather discovered." — George E. Andrews

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