Number Theory: 10 (Dover Books on Mathematics) Paperback – 2 Jan 2000
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About the Author
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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It is essential to do the problems in this book! Do not skip them thinking writing down the definitions and theorems will be enough-- some of the problems will kill you if you go in only knowing the written theorems, without any proper thought into the subject. Like any mathematical subject, it requires rigorous thinking and hours of reading before even considering going on to more advanced topics, like algebraic number theory, abstract algebra, or residue theory.
Breaking down the book into parts, I find it slightly disconcerting that despite the small nature of the book, the concept of quadratic congruences are only introduced in a less-than-introductory fashion, in comparison to other number theory books. It may be true that the author's main research was based off partition theory (the largest section in the book), but quadratic congruences have large applied mathematical influences, and should be considered to be read on, after the book as been finished.
Despite that, this text is an incredible foray into elementary number theory, and is a recommended buy for all those interested in the mathematical world.
Here are the titles of the chapters with their starting pages:
// PART I Multiplicativity-Divisibility // 1. Basis Representation-3 / 2. The Fundamental Theorem of Arithmetic-12 / 3. Combinatorial and Computational Number Theory-30 / 4. Fundamentals of Congruences-49 / 5. Solving Congruences-58 / 6. Arithmetic Functions-75 / 7. Primitive Roots-93 / 8. Prime Numbers-100 // PART II Quadratic Congruences // 9. Quadratic Residues-115 / 10. Distribution of Quadratic Residues-128 // PART III Additivity // 11. Sums of Squares-141 / 12. Elementary Partition Theory-149 / 13. Partition Generating Functions-160 / 14. Partition Identities-175 // PART IV Geometric Number Theory // 15. Lattice Points-201 / There are four mathematical appendices and the full set of indices after the 15 chapters--213-259.
From the complicated table of contents above, one can see a broad sweep of combinatorial number theory. Part I is mostly pretty straight number theory, and that is what I did read. Part III on additivity is almost fully combinatorics more than number theory though. Still the price of this book is quite low to have access to all of this big range of mathematics to pick and choose what is most interesting to any given reader. Recommended.