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Biscuits of Number Theory

Biscuits of Number Theory

Arthur T. Benjamin and Ezra Brown, Editors

Catalog Code: DOL-34
Print ISBN: 978-0-88385-340-5
320 pp., Hardbound, 2009
List Price: $62.50
MAA Member: $49.95
Series: Dolciani Mathematical Expositions

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In Biscuits of Number Theory, the editors have chosen articles that are exceptionally well written and that can be appreciated by anyone who has taken (or is taking) a first course in number theory. This book could be used as a textbook supplement for a number theory course, especially one that requires students to write papers or do outside reading. The editors give examples of some of the possibilities.

The collection is divided into seven chapters: Arithmetic; Primes; Irrationality; Sums of Squares and Polygonal Numbers; Fibonacci Numbers; Number Theoretic Functions; and Elliptic Curves, Cubes, and Fermat's Last Theorem. As with any anthology, you don't have to read the Biscuits in order. Dip into them anywhere: pick something from the Table of Contents that strikes your fancy, and have at it. If the end of an article leaves you wondering what happens next, then by all means dive in and do some research. You just might discover something new!

Table of Contents

Part I: Arithmetic
Part II: Primes
Part III: Irrationality and Continued Fractions
Part IV: Sums of Squares and Polygonal Numbers
Part V: Fibonacci Numbers
Part VI: Number-Theoretic Functions
Part VII: Elliptic Curves, Cubes and Fermat's Last Theorem
About the Editors

Excerpt: Taxicabs and Sums of Two Cubes (p. 259)

Our story begins in 1913, when the distinguished British mathematician G. H. Hardy received a bulky envelope from India full of page after page of equations. Every famous mathematician periodically receives letters from cranks who claim to have proven the most wonderous results. Sometimes the proofs, incorrect or incoherent, are included. At other times the writer solicits a reward in return for revealing his discoveries. Now this letter to Hardy, which was from a poor clerk in Madras by the name of Ramanujan, was filled with equations, all given without any sort of proof. Some of the formulas were well-known, mere exercises; while many of the others looked preposterous to Hardy's trained eye.

Who would have blamed Hardy if he had returned this missive to the sender, unread? And in fact, Ramanujan had previously sent his results to two other British mathematicians, each of whom had done just that! But instead Hardy gave some thought to these "wild theorems. Theorems such as he had never seen before, nor imagined." And together, he and J. E. Littlewood, another eminent mathematician with whom Hardy often worked, succeeded in proving some of Ramanujan's amazing identities. At this point Hardy realized that this letter was from a true mathematical genius, and he became determined that Ramanujan should come to England to pursue his mathematical researches. Using travel money provided by Hardy's college, Ramanujan arrived in 1914. Over the next several years he continued to produce and publish highly original material, and he also collaborated with Hardy on a number of outstanding papers.

In 1918, at the age of 30, Ramanujan was elected a Fellow of the Royal Society and also of Trinity College, both signal honors which he richly deserved. Unfortunately, in the colder climate of England he contracted tuberculosis. He returned to his native Madras and died, in 1920,at the age of 33.

During all of Ramanujan's life, he considered numbers to be his personal friends. To illustrate, Hardy tells the story of how one day he visited Ramanujan in the hospital. At a loss for something to say, Hardy remarked that he had arrived at the hospital in taxicab number 1729. "It seemed to me," he continued, "a rather dull number." To which Ramanujan replied "No, Hardy! It is a very interesting number. It is the smallest number expressible as a sum of two cubes in two different ways:"

1729 = 13 + 123 = 93 + 103.

About the Editors

Ezra (Bud) Brown grew up in New Orleans and has degrees from Rice University and Louisana State University. His doctoral advisor was Gordon Pall, who encouraged him to "write your mathematics as if you wanted someone to read it." Since 1969 he has been in the Mathematics Department at Virginia Tech, where he is currently Alumni Distinguished Professor.

He spent one sabbatical year (and every summer since 1993) in Washington, D. C. and another in Munich. He is the author of some sixty papers, mostly in number theory and discrete mathematics. His translation of Regiomontanus: his life and work, Ernst Zinner's biography of that fifteenth century mathematician/astronomer, was published in 1990 by North-Holland.

He is a member of the Mathematical Assocation of America (MAA) and Pi Mu Epsilon national mathematics honor society. He received the Outstanding Teacher Award from the MD/DC/VA Section of the MAA, and he currently (2008) serves as that section's governor. He received the Carl Allendoerfer Award (2003) and three George Pólya Awards (2000, 2001, 2006) from the MAA for expository writing. He enjoys his family, singing, playing piano, and gardening, and he occasionally bakes biscuits for his students.

Arthur (Art) Benjamin earned his B.S. in Applied Mathematics from Carnegie Mellon and his Ph.D. in Mathematical Sciences from Johns Hopkins. Since 1989, he has taught at Harvey Mudd College, where he is Professor of Mathematics and past Chair. In 2000, he received the Haimo Award for Distinguished Teaching by the Mathematical Association of America. He served as the MAA's Pólya Lecturer from 2006 to 2008.

His research interests include combinatorics and number theory, with a special fondness for Fibonacci numbers. Many of these ideas appear in his book (co-authored with Jennifer Quinn), Proofs that Really Count: The Art of Combinatorial Proof, published by the MAA. In 2006, that book received the Beckenbach Book Prize by the MAA. From 2004 to 2008, professors Benjamin and Quinn were the editors of the Math Horizons magazines, published by the MAA.

Art is also a magician who performs his mixture of math and magic to audiences all over the world, including the Magic Castle in Hollywood. He has demonstrated and explained his calculating talents in his book Secrets of Mental Math and on numerous television and radio programs, including the Today Show, CNN, and National Public Radio. He has bee featured in Scientific American, Omni, Discover, People, Esquire, New York Times, Los Angeles Times, and Reader's Digest. In 2005, Reader's Digest called him "America's best Math Wiz."


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