Biscuits of Number Theory

Transcription

1 The Dolciani Mathematical Expositions NUMBER THIRTY-FOUR Biscuits of Number Theory Edited by ArthurT. Benjamin and Ezra Brown Published and Distributed by The Mathematical Association of America

2 Introduction xi Part I: Arithmetic 1 1. A Dozen Questions About the Powers of Two 3 James Tanton. Math Horizons, vol. 8, no. 1 (September 2001), pp. 5-10; 2002 frevor Evans Award. 2. From 30 to 60 is Not Twice as Hard 13 Michael Dalezman. Mathematics Magazine, vol. 73, no. 2 (April 2000), pp Reducing the Sum of Two Fractions 17 Harris S. Shultz and Ray C. Shiflett. Mathematics Teacher, vol. 98, no. 7 (March 2005), pp A Postmodern View of Fractions and Reciprocals of Fermat Primes Rafe Jones and Jan Pearce. Mathematics Magazine, vol. 73, no. 2 (April 2000), pp ; 2001 Allendoerfer Award. 5. Visible Structures in Number Theory 39 Peter Borwein and Loki Jorgenson. American Mathematical Monthly, vol. 108, no. 10 (December 2001), pp ; 2002 Lester Ford Award. 6. Visual Gems of Number Theory 53 Roger B. Nelsen. Math Horizons, vol. 15, no. 3 (February 2008), pp. 7-9, 31. Part II: Primes A New Proof of Euclid's Theorem 61 Filip Saidak. American Mathematical Monthly, vol. 113, no. 10 (December 2006), pp On the Infinitude of the Primes 63 Harry Furstenberg. American Mathematical Monthly, vol. 62, no. 5 (May 1955), p On the Series of Prime Reciprocals 65 James A. Clarkson. Proceedings oftheams, vol. 17, no. 2 (April 1966), p Applications of a Simple Counting Technique 67 Melvin Hausner. American Mathematical Monthly, vol. 90, no. 2 (February 1983), pp On Weird and Pseudoperfect Numbers 69 S. J. Benkoski and P. Erdos. Mathematics of Computation, vol. 28, no. 126 (April 1974), pp A Heuristic for the Prime Number Theorem 77 Hugh L. Montgromery and Stan Wagon. Mathematical Intelligencer, vol. 28, no. 3 (2006), pp vii

3 13. ATale of Two Sieves ;.. 85 Carl Pomerance. Notices oftheams, (December 1996), pp ; 2001 Conant Prize. Part III: Irrationality and Continued Fractions Irrationality of the Square Root of Two A Geometric Proof 107 Tom M. Apostol. ' American Mathematical Monthly, vol. 107, no. 9 (November 2000), pp Math Bite: Irrationality of Vm 109 Harley Flanders. Mathematics Magazine, vol. 72, no. 3 (June 1999), p A Simple Proof that n is Irrational Ill Ivan Niven. Bulletin oftheams, vol. 53 (1947), p JI, e and Other Irrational Numbers 113 AlanE. Parks. American Mathematical Monthly, vol. 93, no. 9 (November 1986), pp A Short Proof of the Simple Continued Fraction of e 115 Henry Cohn. American Mathematical Monthly, vol. 113, no. 1 (January 2006), pp Diophantine Olympics and World Champions: Polynomials and Primes Down Under 121 Edward B. Burger. American Mathematical Monthly, vol. 107, no. 9 (November 2000), pp ; 2004 Chauvenet Prize. 20. An Elementary Proof of the Wallis Product Formula for Pi 129 Johan Wastlund. American Mathematical Monthly, vol. 114, no. 10 (December 2007), pp The Orchard Problem 133 Ross Honsberger. Mathematical Gems, Chapter 4, pp , Dolciani Mathematical Expositions, MAA, Part IV: Sums of Squares and Polygonal Numbers A One-Sentence Proof that every Prime/? = 1 (mod 4) is a Sum of Two Squares 143 D. Zagier. American Mathematical Monthly, vol. 97, no. 2 (February 1990), p Sum of Squares II 145 Martin Gardner and Dan Kalman. Proofs Without Words: Exercises in Visual Thinking, Classroom Resource Materials, MAA, p Sums of Squares VIII '. 147 Roger B. Nelsen. Proofs Without Words II: More Exercises in Visual Thinking, Classroom Resource Materials, MAA, p. 88. viii

4 25. A Short Proof of Cauchy's Polygonal Number Theorem 149 Melvyn B. Nathanson. Proceedings of the AMS, vol. 99, no. 1 (January 1987), pp Genealogy of Pythagorean Triads 153 A. Hall. Mathematical Gazette, vol. 54, no. 390 (December 1970), pp Part V: Fibonacci Numbers A Dozen Questions About Fibonacci Numbers 157 James Tanton. Math Horizons, vol. 12, no. 3 (February 2005), pp The Fibonacci Numbers Exposed 167 Dan Kalman and Robert Mena. Mathematics Magazine, vol. 76, no. 3 (June 2003), pp The Fibonacci Numbers Exposed More Discretely 183 Arthur T. Benjamin and Jennifer J. Quinn. Mathematics Magazine, vol. 76, no. 3 (June 2003), pp Part VI: Number-Theoretic Functions Great Moments of the Riemann zeta Function 199 Jennifer Beineke and Chris Hughes. Original article. 31. The Collatz Chameleon 217 Marc Chamberland. Math Horizons, vol. 14, no. 2 (November 2006), pp Bijecting Euler's Partition Recurrence 223 David M. Bressoud and Doron Zeilberger. American Mathematical Monthly, vol. 92, no. 1 (January 1985), pp Discovery of a Most Extraordinary Law of the Numbers Concerning the Sum of Their Divisors 225 Leonard Euler. Translated by George Polya. Mathematics and Plausible Reasonings Volume I, Princeton University Press, (1954), pp The Factorial Function and Generalizations 233 Manjul Bhargava. American Mathematical Monthly, vol. 107, no. 9 (November 2000), pp ; 2003 Hasse Prize. 35. An Elementary Proof of the Quadratic Reciprocity Law 251 Sey Y.Kim. American Mathematical Monthly, vol. Ill, no. 1 (January 2004), pp Part VII: Elliptic Curves, Cubes and Fermat's Last Theorem Proof Without Words: Cubes and Squares 257 J. Barry Love. Mathematics Magazine, vol. 50, no. 2 (March 1977), p. 74. IX

5 37. Taxicabs and Sums of Two Cubes 259 Joseph H. Silverman. American Mathematical Monthly, vol. 100, no. 4 (April 1993), pp ; 1994 Lester Ford Award. 38. Three Fermat Trails to Elliptic Curves 273 Ezra Brown. The College Mathematics Journal, vol. 31, no. 3 (May 2000), pp ; 2001 Polya Award. 39. Fermat's Last Theorem in Combinatorial Form 285 W.V. Quine. American Mathematical Monthly, vol. 95, no. 7 (August-September 1988), p "A Marvelous Proof 287 Fernando Q. Gouvea. American Mathematical Monthly, vol. 101, no. 3 (March 1994), pp ; 1995 Lester Ford Award. About the Editors 311