9.3 Number theory (also see 0.1) The Irrationality of Certain Numbers, Peter A. Lindstrom, 1:1, 1970, 30-31, 0.2 F(1) Rejection Theorem, Howard Sarr, 1:2, 1970, 39-40 F(1) and F(d) Rejection Theorems, William I. Miller, 2:2, 1971, 95-96 Pythagorean Triples by Geometry, Steven L. Kleiman, 3:1, 1972, 39-41 Anomalous Cancellation, R. P. Boas, Jr., 3:2, 1972, 21-24 ab=c, Sidney Penner, 4:2, 1973, 86-87, C Fermat Numbers, W. G. Leavitt, 4:3, 1973, 7-10 Random Sieving and the Prime Number Theorem, Karl Greger, 5:1, 1974, 41-46, 5.3.2 The Computer as an Aid to Discovery, Frederick H. Young, 5:3, 1974, 55-57 On Generalized h-Base, Norman Woo, 6:3, 1975, 16-17 Quasi-Pythagorean Triples for an Oblique Triangle, Kay Dundas, 8:3, 1977, 152-155, 0.6 Methods of Random Number Generation, Edwin G. Landauer, 8:5, 1977, 296-303 A Note on Angle Construction, Richard L. Francis, 9:2, 1978, 73-75 The Pigeonhole Principle, Kenneth R. Rebman, 10:1, 1979, 3-13, 3.1 Triangular Squares, Bill Leonard and Harris S. Schultz, 10:3, 1979, 169-171 Two Distinguished Integers, Ross Honsberger, 10:3, 1979, 195-197 Billiard Balls and a Number Theory Result, Charles H. Jepsen, 10:5, 1979, 306-312 The Use of Generating Functions to Discover and Prove Partition Identities, Henry L. Alder, 10:5, 1979, 318-329 On Sets of Points in the Plane and A Property of the Binomial Coefficients, Ross Honsberger, 11:2, 1980, 116-119, 0.3 A Combinatorial Proof of Euler's Formula, Iain T. Adamson, 11:4, 1980, 272-273, C, 3.2 Another Derivation of a Double Inequality, Norman Schaumberger, 11:4, 1980, 273, C An Elementary Gem Concerning pi(n), the Number of Primes less than or equal to n, Ross Honsberger, 11:5, 1980, 305-312 Factoring Factorials, Richard J. Friedlander, 12:1, 1981, 12-20 A Geometric Motivation of Fermat's Factoring Method, Michael Ratliff, 12:1, 1981, 24-27 Short Stories in Number Theory, Ross Honsberger, 12:1, 1981, 34-40 Some Conjectures on Fermat's Last Conjecture, Lawrence Sher and David Sher, 12:1, 1981, 51-52, C Applying Complex Arithmetic, Herbert L. Holden, 12:3, 1981, 190-194, 0.6, 5.3.1, 9.5 Short Stories in Number TheoryŃPart II, Ross Honsberger, 12:4, 1981, 280-282 Forward and Backward with Euclid, Gary E. Stevens, 12:5, 1981, 302-306 How Many Positive Integers Have Nines in Their Decimal Representations?, Calvin T. Long, 12:5, 1981, 320-324 Short Stories in Number TheoryŃPart III, Ross Honsberger, 12:5, 1981, 325-329 A Classroom Approach to x^2 + y^2 + z^2 = w^2, Norman Schaumberger, 12:5, 1981, 331-332, C, 0.4 Synthetic Division Shortened, Warren Page and Leo Chosid, 12:5, 1981, 334-336, C, 0.7 Smith Numbers, A. Wilansky, 13:1, 1982, 21, 0.1 Semi-Regular Lattice Polygons, Ross Honsberger, 13:1, 1982, 36-44, 3.1 A Simple Divisibility Algorithm, David Y. Hsu, 13:1, 1982, 58-59, C, 0.2 Remark on an Elementary Gem Concerning PI(n), Branislav Martic, 13:2, 1982, 158-159, C Sums of Powers of the First n Integers, David Y. Hsu, 13:3, 1982, 196-197, C Representable Integers, Ross Honsberger, 13:4, 1982, 260-265 Isomorphisms on Magic Squares, Ali R. Amir-Moez, 14:1, 1983, 48-51, 0.2, 5.4.1, 9.2, 9.4 A Prime-Generating Function, Donald D. Elliot, 14:1, 1983, 57, C The Alluring Lore of Cyclic Numbers, Michael W. Ecker, 14:2, 1983, 105-109 License Numbers and Divisibility Rules, Harry Hutchins, 14:2, 1983, 122-125 Minimization Based on the Greatest Common Divisor, David Y. Hsu, 14:2, 1983, 165-166, C Congruences of Cyclotomic Polynomials, Phyllis Lefton, 14:3, 1983, 257-258, C SSD Persistence: A Mathematical System for Student Investigation, John Scheding, 14:4, 1983, 309-312, 1.2 A Tiling of the Plane with Triangles, Paul T. Mielke, 14:5, 1983, 377-381, 0.3, 9.2 The Address Problem, Michael Tennor, 14:5, 1983, 407-414, 0.2 Digital Roots of Mersenne Primes and Even Perfect Numbers, Syed Asadulla, 15:1, 1984, 53-54, C Integer-Sided Triangles with One Angle Twice Another, R. S. Luthar, 15:1, 1984, 55-56, C, 0.6 The Distribution of First Digits, Stephen H. Friedberg, 15:2, 1984, 120-125, 7.2 Repeating Decimals, W. G. Leavitt, 15:4, 1984, 299-308 On the Natural Density of the Niven Numbers, Robert E. Kennedy and Curtis N. Cooper, 15:4, 1984, 309-312, 7.3 Pythagorean Systems of Numbers, Joseph Wiener, 15:4, 1984, 324-326, C, 0.2, 0.4 An Approach to Problem-Solving Using Equivalence Classes Modulo n, James E. Schultz and William F. Burger, 15:5, 1984, 401-405, 0.2 The Computation of Repeating Decimals, T. E. Ganter, 15:5, 1984, 436-440 What Do I Know? A Study of Mathematical Self-Awareness, Philip J. Davis, 16:1, 1985, 22-41, 0.2 Generalized Pythagorean Triples, W. J. Hildebrand, 16:1, 1985, 48-52, 0.6, 5.5 Medical Cozenage on Fermat's Last Theorem, Lee Whitt, 16:1, 1985, 55-56, C The House Number Problem and its Variations, Joey Paul, 16:2, 1985, 108-117 A New Divisibility Algorithm, Joseph Whittaker, 16:4, 1985, 268-276, 0.2 The International Mathematical Olympiad Training Session, Cecil Rousseau and Gregg Patruno, 16:5, 1985, 362-365, 0.3, 2.2 Computing Large Factorials, Gerard Kiernan, 16:5, 1985, 403-412, 9.6 Angling for Pythagorean Triples, Dan Kalman, 17:2, 1986, 167-168, C, 0.4 From None to Infinity: Challenging Problems in Cardinality Classification, Richard L. Francis, 17:3, 1986, 226-230 The Distribution of First j Digits, S. A. Patil and V. R. R. Uppuluri, 17:3, 1986, 240-243, C Cryptology: From Ceasar Ciphers to Public-Key Cryptosystems, Dennis Luciano and Gordon Prichett, 18:1, 1987, 2-17, 7.2, 0.1 Bach, 5465, and Upside-Down Numbers, Robert E. Kennedy and Curtis N. Cooper, 18:2, 1987, 111-115 Generating Functions, William Watkins, 18:3, 1987, 195-211, 6.3, 5.4.2 The Chinese Remainder Problem and Polynomial Interpolation, Isaac J. Schoenberg, 18:4, 1987, 320-322, C On Partitioning a Real Number, William Staton, 19:1, 1988, 53-54, C, 5.1.4 Mathematical Haystacks: Another Look at Repunit Numbers, Richard L. Francis, 19:3, 1988, 240-246 Involutions and Problems Involving Perimeters and Area, Joseph Wiener and Henjin Chi and Hushang Poorkarimi, 19:3, 1988, 250-252, C, 9.5 Sieving Primes on a Micro, Harley Flanders and Alan F. Tomala, 19:4, 1988, 364-367, 8.1 Amalgamation fo Formulae for Sequences, N. S. Mendelsohn, 19:5, 1988, 421-424, C Pseudorandom Number Generators and a Four-Bit Computer System, James C. Reber, 20:1, 1989, 54-55, C, 6.3, 9.10 Finding Rational Roots of Polynomials, Don Redmond, 20:2, 1989, 139-141, C, 0.7 It's Magic! Multiplication Theorems for Magic Squares, Daniel Widdis and R. Bruce Richter, 20:4, 1989, 301-306, 3.2, 9.2 Locating Multiples of Primes in Pascal's Triangle, Lawrence O. Cannon, 20:4, 1989, 324-328, C Strings of Strongly Composite Integers and Invisible Lattice Points, Peter Schumer, 21:1, 1990, 37-40, C Computer-Aided or Analytic Proof?, Herve Lehning, 21:3, 1990, 228-239 Student Research Projects: Self-esteem in Mathematics, Herbert S. Wilf, 21:4, 1990, 274-277, 1.2 Triangles with Integer Sides and Sharing Barrels, David Singmaster, 21:4, 1990, 278-285, 0.4 The Birth of the Eotvos Competition, Agnes Arvai Wieschenberg, 21:4, 1990, 286-293, 2.2 Polar Summation, Loretta McCarty, 21:5, 1990, 397-398, C Another Proof of the Irrationality of the Square Root of 2, Enzo R. Gentile, 22:2, 1991, 143, C Secrets of the Faro: Student Research Project, Irl C. Bivens, 22:2, 1991, 144-147, 9.4 The Mathematics of Identification Numbers, Joseph A. Gallian, 22:3, 1991, 194-202, 9.4 Reward of the Rings: Student Research Projects, Irl C. Bivens, 22:5, 1991, 418-420, 9.4 Summation by Parts, Gregory Fredricks and Roger B. Nelsen, 23:1, 1992, 39-44, C, 5.1.2, 5.4.1, 5.4.2 The Probability that (a, b)=1, Aaron D. Abrams and Matteo J. Paris, 23:1, 1992, 47, C Number Theory and Linear Algebra: Exact Solutions of Integer Systems, George Mackiw, 23:1, 1992, 52-58, 4.1 A Serendipitous Application of the Pythagorean Triplets, Susan Forman, 23:4, 1992, 312-314, C, 0.2 Primitive Pythagorean Triples: Student Research Project, Ernest J. Eckert, 23:5, 1992, 413-417 Sums of Triangular Numbers, Roger B. Nelsen, 23:5, 1992, 417, C Geometry: A Gateway to Understanding, Peter Hilton and Jean Pedersen, 24:4, 1993, 298-317, 0.3 Towers of Powers Modulo m, Robert J. MacG. Dawson, 25:1, 1994, 22-28 Eisenstein's Misunderstood Geometric Proof of the Quadratic Reciprocity Theorem, Reinhard C. Laubenbacher and David J. Pengelley, 25:1, 1994, 29-34 Frequencies of Digits in Factorials: An Experimental Approach, Michael L. Treuden, 25:1, 1994, 48-55 Euclid's (Gaussian) Algorithm: A Lattice Approach, Steve Benson, 25:2, 1994, 118-124 Approaches to the Formula for the nth Fibonacci Number, Russell Jay Hendel, 25:2, 1994, 139-142, C, 0.2, 4.5, 5.4.2, 9.5 Sums of Odd Squares, Roger B. Nelsen, 25:3, 1994, 246, C Prime Number Records, Paulo Ribenboim, 25:4, 1994, 280-290 Investigation of a Recurrence Relation: Student Research Project, Dmitri Thoro and Linda Valdes, 25:4, 1994, 322-324, 3.2, 6.3 A Mathematica'l Magic Trick, Stan Wagon, 25:4, 1994, 325-326, C FFF #79. A Divisibility Property, Ed Barbeau, 25:5, 1994, 433, F FFF #82. Why Wiles' Proof of the Fermat Conjecture is False, Ed Barbeau, 25:5, 1994, 434-435, F, 9.7 The Repeating Integer Paradox, Paul Fjelstad, 26:1, 1995, 11-15 A Taylor-made Plug for Wiles' Proof, Nigel Boston, 26:2, 1995, 100-105 More Mathematical Gems, Ross A. Honsberger, 26:4, 1995, 281-283, 9.5 A Surprise Regarding the Equation phi(x) = 2(6n+1), Joseph B. Dence and Thomas P. Dence, 26:4, 1995, 297-301 Exploring Fibonacci Numbers Mod M, Jack Ryder, 27:2, 1996, 122-124, C, 3.3 The Square of Any Odd Number is the Difference Between Two Triangular Numbers (Proof Without Words), Roger B. Nelsen, 27:2, 1996, 118, C, 0.1 Fractions with Cycling Digit Patterns, Dan Kalman, 27:2, 1996, 109-115, 0.1 Pythagorean Triples: The Hyperbolic View, Raymond A. Beauregard and E. R. Suryanarayan, 27:3, 1996, 170-181, 9.4 FFF #108. All Perfect Numbers Are Even, Ari Turner, 27:4, 1996, 283, F Generalizations of a Mathematical Olympiad Problem, Joe Klerlein and Scott Sportsman, 27:4, 1996, 296-297, 3.2 Three Applications of a Familiar Formula, Robert A. Fontenot, 27:5, 1996, 356-360 Periodic Points of the Difference Operator, Chris Bernhardt and Thomas Yuster, 2:1, 1997, 20-26 Digital Permutations, Bryan Dawson, 28:1, 1997, 26, C A Long Sequence of Composite Numbers, Ed Pegg, Jr., 28:2, 1997, 121, C Fibonacci Powers and a Fascinating Triangle, Dale K. Hathaway and Stephen L. Brown, 28:2, 1997, 124-128, C, 3.3, 6.3 Two Identities for Triangular Numbers (proof by picture), Roger B. Nelsen, 28:3, 1997, 197, C On Dividing Coconuts: A Linear Diophantine Problem, Sahib Singh and Dip Bhattacharya, 28:3, 1997, 203-204, C, 5.4.3 Are There Functions That Generate Prime Numbers?, Paulo Ribenboim, 28:5, 1997, 352-359 The Brahmagupta Triangles, Raymond A. Beauregard and E. R. Surynarayan, 29:1, 1998, 13-17, 0.4 A Class of Pleasing Periodic Designs, Federico Fernandez, 29:1, 1998, 18-26, 4.3, 9.4 Making Squares from Pythagorean Triangles, Charles Jepsen and Roc Yang, 29:4, 1998, 284-288, 9.7 On Factoring n with the b-algorithm, Vincent Lucarelli, 29:4, 1998, 289-295 Egyptian Fractions and the Inheritance Problem, Premchand Anne, 29:4, 1998, 296-300 More Coconuts, Sidney H. Kung, 29:4, 1998, 312-313, C, 0.1 Square Roots From 1;24,51,10 to Dan Shanks, Ezra Brown, 30:2, 1999, 82-95 From Euler to Fermat, Hidefumi Katsuura, 30:2, 1999, 118-119, 9.5 Palindromic Primes, Harvey Dubner, 30:4, 1999, 292, C Powers as Uniform Sums of Positive Squares, Robert J. Wisner, 30:4, 1999, 293-296 Progress on the Tarry-Escott-Prouhet Problem, the editor, 31:1, 2000, 68, C Recursions That Produce Pythagorean Triples, Peter W. Wade and William R. Wade, 31:2, 2000, 98-101 General Arithmetic Triangles and BhaskaraÕs Equation, Raymond Beauregard and E. R. Surynarayan, 31:2, 2000, 111-115 Three Fermat Trails to Elliptic Curves, Ezra Brown, 31:3, 2000, 162-172 Meta-Problems in Mathematics, Al Cuoco, 31:5, 2000, 373-378, 0.7, 5.1.2 A Polynomial with a Root Mod m for Every m, Allen J. Schwenk, 31:5, 2000, 403-405, C, 9.4 The Lord Over Better and Worse Births, John Fossa and Glenn Erickson, 32:3, 2001, 185-193, 9.2 Magic Squares, Finite Planes, and Points of Inflection on Elliptic Curves, Ezra Brown, 32:4, 2001, 260-267, 5.1.3, 9.2 Powers Made Easy, James Kirby, 32:5, 2001, 329, C, 0.1 Close!, Noam Elkies, 33:1, 2002, 16, C A Visit With Six, Monte J. Zerger, 33:2, 2002, 74-87, 9.2 ItÕs Perfectly Rational, Philip K. Hotchkiss, 33:2, 2002, 113-117, 5.1.4 A Ramanujan Result Viewed From Matrix Algebra, Raymond A. Beauregard and E. R. Suryanarayan, 33:3, 2002, 212-214, 4.1, 9.4 FermatÕs Little Theorem From the Multinomial Theorem, Thomas J. Osler, 33:3, 2002, 239, C A Generalized Chinese Remainder Theorem, Fredric T. Howard, 33:4, 2002, 279-282 A Numerical Introduction to Partial Fractions, Eric L. McDowell, 33:5, 2002, 400-403, C, 5.2.4 A Magic Trick from Fibonacci, James Smoak and Thomas J. Osler, 34:1, 2003, 58-60, C Recursive Enumeration of Pythagorean Triples, Darryl McCullough and Elizabeth Wade, 34:2, 2003, 107-111 Rational Boxes, Sidney Kung, 34:3, 2003, 182, C, 5.1.4 Coin ToGa: A Coin-Tossing Game, Osvaldo Marrero and Paul C. Pasles, 34:3, 2003, 183-193, 7.2 Variations on a Theme from PascalÕs Triangle, Thomas J. Osler, 34:3, 2003, 216-223 Partitioning Triangular Numbers, Matthew Haines and Michael Jones, 34:4, 2003, 295, C A large square consisting only of digits 7, 8 and 9, Hisanori Mishima, 34:4, 2003, 303, C, 0.1 On a Diophantine Equation and its Ramifications, Titu Andreescu and Dorin Andrica, 35:1, 2004, 15-21 MidyÕs (Nearly) Secret Theorem Š An Extension After 165 Years, Brian D. Ginsberg, 35:1, 2004, 26-30 Five Mathematicians, a Bunch of Coconuts, a Monkey, and a Coin, John E. Morrill, 35:4, 2004, 256-257 On a Three-Dimensional Generalization of FermatÕs Area Theorem, Raymond A. Beauregard and Konstantine D. Zelator, 35:4, 2004, 289-291 Discovering Roots: Ancient, Medieval, and Serendipitous, Bryan Dorner, 36:1, 2005, 35-43, 0.2, 2.1, 4.5 Irrational Roots of Integers, Ayshhyah Khazad and Allen J. Schwenk, 36:1, 2005, 56-57, C (see also 36:4, 317) An Upper Bound on the nth Prime, John H. Jaroma, 36:2, 2005, 158-159, C M&m Sequences, Harris S. Shultz and Ray C. Shiflett, 36:3, 2005, 191-198, 6.3 On Sums of Cubes, Hajrudin Fejzic, Dan Rinne, and Bob Stein, 36:3, 2005, 226-228, C Curious Consequences of a Miscopied Quadratic, Jeffrey L. Poet and Donald L. Vestal, Jr., 36:4, 2005, 273-277 On Primes, Density Measures, and Statistical Independence, Yung-Pin Chen, 36:4, 2005, 284-288, 7.2 A Perplexing Polynomial Puzzle, Revisited, Folkmar Bernemann and Stan Wagon, 36:4, 2005, 288, C Visibles Revisited, Mark Bridger and Andrei Zelevinsky, 36:4, 2005, 289-300 FFF #241. A triangle condition, Ed Barbeau, 36:4, 2005, 315-316, F (see also Ken McCaffrey, 37:3, 2006, 215-216, F) A Variant of the Partition Function, John F. Loase, David Lansing, Cassie Hryczaniuk, and Jamie Cahoon, 36:4, 2005, 320-321, C Exactly When Is (a+b)^n equivalent to a^n + b^n (mod n)?, Pratibha Ghatage and Brian Scott, 36:4, 2005, 322, C RamanujanÕs Continued Fraction for a Puzzle, Poo-Sung Park, 36:5, 2005, 363-365 (Errata on 37:5, 2006, 369) A Paper-and-Pencil gcd Algorithm for Gaussian Integers, Sandor Szabo, 36:5, 2005, 374-380, 9.4 A Two-Parameter Trigonometry Series, Xiang-Qian Chang, 36:5, 2005, 408-412, C, 9.5 Using Random Tilings to Derive a Fibonacci Congruence, Keith Neu and Paul Deiermann, 37:1, 2006, 44-47, C Parity and Primality of Catalan Numbers, Thomas Koshy and Mohammad Salmassi, 37:1, 2006, 52-53, C, 3.2 Student Research Project: Integer Points on a Hyperboloid of One Sheet, Margaret Beattie and Chester Weatherby, 37:1, 2006, 54-58, C No Arithmetic Cyclic Quadrilaterals, Raymond A. Beauregard, 37:2, 2006, 110-113 Searching for Mobius, Al Cuoco, 37:2, 2006, 137-142, C Where are the zeros of zeta of s? (poem), Tom M. Apostol, 37:2, 2006, 163, C What Tom Apostol DidnÕt Know (poem), Saunders MacLane, 37:2, 2006, 164, C Fibonacci Identities via the Determinant Sum Property, Michael Z. Spivey, 37:4, 2006, 286-289, 3.2, 4.2 FFF. Sums of 12th powers, Ed Barbeau, 37:4, 2006, 292, F More Designer Decimals: The Integers and Their Geometric Extensions, O-Yeat Chan and Jim Smoak, 37:5, 2006, 355-363 FFF #260. Increasing a square to a square, Chris Fisher, 38:1, 2007, 43, F, 0.2 Freaky fractions, Rick Kreminsky, 38:1, 2007, 46, C, 0.1 Fibonacci-Like Sequences and Pell Equations, Ayoub B. Ayoub, 38:1, 2007, 49-53, C Sums of Consecutive Integers, Wai Yan Pong, 38:2, 2007, 119-123 Pythagorean Triples with Square and Triangular Sides, Sharon Brueggeman, 38:2, 2007, 138-140, C Surprising Connections between Partitions and Divisors, Thomas J. Osler, Abdulkadir Hassan, and Tirupathi R. Chandrupatla, 38:4, 2007, 278-287 Student Research Project: From Cyclic Sums to Projective Planes, Roger Zarnowski, 38:4, 2007, 304-308, 9.7 Partial Fractions in Calculus, Number Theory, and Algebra, C. A. Yackel and J. K. Denny, 38:5, 2007, 362-374, 5.2.4, 9.4 Summing Up the Euler phi Function, Paul Loomis, Michael Plytage, and John Polhill, 39:1, 2008, 34-42 A Quick Change of Base Algorithm for Fractions, Juan B. Gil and Michael D. Weiner, 39:1, 2008, 56-59, C A New Property of Repeating Decimals, Jane Arledge and Sarah Tekansik, 39:2, 2008, 107-111 Remainder Wheels and Group Theory, Lawrence Brenton, 39:2, 2008, 129-135, 0.1, 9.4 On the Number of Trailing Zeros in n!, David S. Hart, James E. Marengo, Darren A. Narayan and David S. Ross, 39:2, 2008, 139-141, C Centaurs: Here, There, Everywhere!, Dimitri Dziabenko and Oleg Ivrii, 39:4, 2008, 267-272, 6.3, 9.5 Fetching Water with Least Residues, Herb Bailey, 39:4, 2008, 304-306, C, 9.2 Leftist Numbers, Andrew Rich, 39:5, 2008, 330-336, 9.4 Report from the Ambassador to Cida-2, Clifton Cunningham, 39:5, 2008, 337-345, 9.5 An Elementary Trigonometric Equation, Victor H. Moll, 39:5, 2008, 395-399, C, 0.6