Wonders of Numbers

Adventures in Math, Mind, and Meaning

Clifford A. Pickover

Oxford University Press, 2000

"The prolific Pickover dazzles us once more with his book Wonders of Numbers. This big book of mathematical ideas provides hours, days, months if not years of entertaining numbers, puzzles, problems and novelties to explore. The book runs the gamut with such things as X-File numbers, Mozart numbers, Katydid sequences, and on and on. There's something for everyone's interest."
- Theoni Pappas, author of The Joy of Mathematics
"Clifford Pickover has written another marvelous book. Through conversations between whimsical Dr. Googol and his pupil Monica, you can test your wits on an incredible variety of unusual mathematical puzzles and games. Along the way there are fascinating historical facts and math gossip to enjoy. You can't help absorbing a great deal of important math as you pick your way through Pickover's delightful pick of fresh, little-known gems of recreational mathematics."
- Martin
Click here to see the book at Amazon.Com!

Who are the eight most influential female mathematicians? Why aren't Roman numerals used anymore? Why was the first woman mathematician brutally murdered? What were the Unabomber's ten most mathematical technical papers?

Prepare yourself for a shattering odyssey as Wonders of Numbers unlocks the doors of your imagination. The thought-provoking mysteries, puzzles, and problems range from the most beautiful formula of Ramanujan (India's most famous mathematician) to the Leviathan number, a number so big that it makes a trillion pale in comparison. The mysterious puzzles and games should cause even the most left-brained readers to fall in love with numbers. The quirky and exclusive surveys on mathematicians' lives, scandals, and passions will entertain people at all levels of mathematical sophistication.

Grab a pencil. Relax. Then take off on a mind-boggling journey to the ultimate frontier of math, mind, and meaning, as acclaimed author Dr. Clifford Pickover and legendary, eccentric mathematician Dr. Francis Googol explore some of the oddest and quirkiest highways and byways of the numerically obsessed. With numerous illustrations and appendices allowing computer explorations, this is an original, fun-filled, and thoroughly unique introduction to numbers and their role in creativity, computers, games, practical research, and absurd adventures that teeter on the edge of logic and insanity.

Sample program code and color images

Acknowledgments
A Word From the Publisher About Dr. Googol
Preface: One Fish, Two Fish, and Beyond...

Part I. Fun Puzzles and Quick Thoughts

1   Attack of the Amateurs (Sample chapter)
2   Why Don't We Use Roman Numerals Anymore?
3   In a Casino
4   The Ultimate Bible Code
5   How Much Blood?
6   Where are the Ants?  
7   Spidery Math
8   Lost in Hyperspace
9   Along Came a Spider ! 
10  Numbers Beyond Imagination
11  Flatworm Math
12  Cupid's Arrow
13  Poseidon Arrays
14  Scales of Justice
15  Mystery Squares
16  Quincunx
17  Jerusalem Overdrive 
18  The Pipes of Papua
19  The Fractal Society
20  The Triangle Cycle
21  IQ-Block
22  Riffraff
23  Klingon Paths
24  Ouroborous Autophagy
25  Interview With A Number
26  The Dream-Worms of Atlantis
27  Satanic Cycles
28  Persistence
29  Hallucinogenic Highways

Part II. Quirky Questions, Lists, and Surveys

30  Why Was the First Woman Mathematician Murdered?
31  What If We Receive Messages From the Stars?
32  A Ranking of the Four Strangest Mathematicians Who Ever Lived
33  Einstein, Ramanujan, Hawking
34  A Ranking of the Eight Most Influential Female Mathematicians
35  A Ranking of the Five Saddest Mathematical Scandals
36  The Ten Most Important Unsolved Mathematical Problems
37  A Ranking of  the Ten Most Influential Mathematicians Who Ever Lived
38  What is  Gödel's Mathematical Proof of the Existence of God?
39  A Ranking of the Ten Most Influential Mathematicians Alive Today
40  A Ranking of the Ten Most Interesting Numbers
41  The Unabomber's Ten Most Mathematical Technical Papers
42  The Ten Mathematical Formulas the Changed the Face of the World
43  The Ten Most Difficult-to-Understand Areas of Mathematics
44  The Ten Strangest Mathematical Titles Ever Published
45  The 15 Most Famous Transcendental Numbers
46  What is Numerical Obsessive-Compulsive Disorder
47  Who is the Number King?
48  What One Question Would You Add?
49  Cube Maze

Part III. Fiendishly Difficult Digital Delights

50 Hailstone Numbers
51 The Spring of Khosrow Carpet
52 The Omega Prism
53 The Hunt For Double Smoothly Undulating Integers
54 Alien Snow: A Tour of Checkerboard Worlds
55 Beauty, Symmetry and Pascal's Triangle
56 Audioactive Decay
57 Dr. Googol's Prime Plaid
58 Saippuakauppias
59 Emordnilap Numbers
60 The Dudley Triangle
61 Mozart Numbers
62 Hyperspace Prisons
63 Triangular Numbers
64 Hexagonal Cats
65 The X-Files Number
66 A Low-Calorie Treat
67 óõõóóóõõõõ 
68 The Hunt for Elusive Squarions
69 Arranging Alien Heads
70 Katydid Sequences
71 Pentagonal Pie
72 An A?
73 Beauty and the Bits
74 Mr. Fibonacci's Neighborhood
75 Juggler Numbers
76 Apocalypse Numbers
77 The Wonderful Emirp, 1597
78 The Big Brain of Brahmagupta
79 1001 Scheherazades
80 73,939,133
81 5-Numbers from Los Alamos
82 Creator Numbers b
83 Princeton Numbers
84 Parasite Numbers
85 Madonna's Number Sequence
86 Apocalyptic Powers
87 The Leviathan Number _
88 The Safford Number: 365,365,365,365,365,365
89 The Aliens from Independence Day
90 One Decillion Cheerios
91 Undulation in Monaco
92 The Latest Gossip on Narcissistic Numbers
93 The abcdefghij Problem
94 Grenade Stacking
95 The 450-Pound Problem
96 The Hunt For Primes in Pi
97 Schizophrenic Numbers
98 Perfect, Amicable, and Sublime Numbers
99 Prime Cycles and â
100 Cards, Frogs, and Fractal Sequences
101 Fractal Checkers
102 Doughnut Loops
103 Everything You Wanted to Know About Triangles But Were Afraid to Ask
104 Cavern Genesis as a Self-Organizing System
105 Magic Squares, Tesseracts, and Other Oddities
106 Fabergé Eggs Synthesis
107 Beauty and Gaussian Rational Numbers
108 A Brief History of Smith Numbers
109 Alien Ice Cream

Part IV. The Peruvian Collection

110 The Huascarán Box
111 The Intergalactic Zoo
112 The Lobsterman From Lima
113 The Incan Tablets
114 Chinchilla Overdrive
115 Peruvian Laser Battle
116 The Emerald Gambit
117 Wise Viracocha
118 Zoologic
119 Andromeda Incident
120 Yin or Yang
121 A Knotty Challenge at Tacna
122 An Incident at Chavín de Huántar
123 An Odd Symmetry
124 The Monolith at Madre de Dios
125 Amazon Dissection
126 Three Weird Problems with Three
127 Zen Archery
128 Treadmills and Gears
129 Anchovy Marriage Test

Further Exploring
Smorgasbord for Computer Junkies
Further Reading

Visit Oxford University Press's Wonders of Numbers Web Site

Color Images and Program Codes: Oxford University Press is delighted to provide a web site that contains a smorgasbord of computer program listings for Clifford A. Pickover's Wonders of Numbers. Readers have often requested on-line code that they can study and with which they may easily experiment. We hope the code clarifies some of the concepts discussed in Wonders of Numbers. See the chapters in the book for additional explanations.

Color images are also provided here for several of the black and white figures in the book.

Sample Chapter, Wonders of Numbers, by Cliff Pickover

Chapter 1. Attack of the Amateurs

Every productive research scientist cultivates and relies upon nonrational processes to direct his or her own creative thinking. Watson and Crick used visualization to conceive the DNA molecule's configuration. Einstein used visualization to imagine riding on a light beam. Mathematician Ramanujan usually saw a vision of his family Goddess Narnagiri whenever he conceived of a new mathematical formula. The heart of good science is the harmonious integration of good luck in making uncommonly made observations, nonrational processes that are only poorly suggested by the words "creativity" and "intuition." - John Waters, Skeptical Inquirer

Amazingly, lack of formal education can be an advantage. We get stuck in our old ways. Sometimes, progress is made when someone from the outside looks at mathematics with new eyes. - Doris Schattschneider, Los Angeles Times

Are you a mathematical amateur? Do not fret. Many amazing mathematical findings have been made by amateurs, from homemakers to lawyers. These amateurs developed new ways to look at problems that stumped the experts.

Have any of you seen the movie Good Will Hunting in which 20-year-old Will Hunting survives in his rough, working-class, South Boston neighborhood? Like his friends, Hunting does menial jobs between stints at the local bar and run-ins with the law. He's never been to college, except to scrub floors as a janitor at MIT. Yet he can summon obscure historical references from his photographic memory, and almost instantly solve math problems that frustrate the most brilliant professors.

This is not as far-fetched as it sounds! Although you might think that new mathematical studies can only be made by professors with years of training, beginners have also made substantial contributions. Here are some of Dr. Googol's favorite examples:

In the 1970s, Marjorie Rice, a San Diego housewife and mother of five, was working at her kitchen table when she discovered numerous new geometrical patterns that professors had thought were impossible. Rice had no training beyond high school, but by 1976 she had discovered 58 special kinds of pentagonal tiles, most of them previously unknown. Her most advanced diploma was a 1939 high school degree for which she had taken only one general math course. The moral to the story? It's never too late to enter fields and make new discoveries. Another moral: Never underestimate your mother!

In 1998, college student Roland Clarkson discovered the largest prime number known at the time. (A prime number like 13 is evenly divisible only by 1 and itself.) The number was so large that it could fill several books. In fact, some of the largest prime numbers these days are found by college students using a network of cooperating personal computers and software downloadable from the Internet.

In the early 1600s, Pierre de Fermat, a French lawyer, made brilliant discoveries in number theory. Although he was an "amateur" mathematician, he created mathematical puzzles such as "Fermat's Last Theorem" which was not solved until 1994. Fermat was no ordinary lawyer indeed. He is considered, along with Blaise Pascal, as the founder of probability theory. As the coinventor of analytic geometry, he is considered, along with René Descartes, as one of the first modern mathematicians.

In the mid-1990s, Texas banker Andrew Beal posed a perplexing mathematical problem and offered $5,000 for the solution of this problem. The value of the prize increases by $5,000 per year up to $50,000 until it is solved. In particular, Beal was curious about the equation A^x + B^y = C^z. The six letters represent integers, with x, y, and z greater than 2. (Fermat's Last Theorem involves the special case in which the exponents x, y, and z are the same.) Oddly enough, Beal noticed that when a solution of this general equation existed, then A, B, and C have a common factor. For example, in the equation 3⁶ + 18³ = 3⁸, the numbers 3, 18, and 3 all have the factor 3. Using computers at his bank, Beal checked equations with exponents up to 100 but could not discover a solution that didn't involve a common factor. He wondered if this is always true. R. Daniel Mauldin of the University of North Texas commented in the December, 1997 Notices of the American Mathematical Society, "It is remarkable that occasionally someone working in isolation, and with no connections to the mathematical community, formulates a problem so close to current research activity."

In 1998, 17-year-old Colin Percival, calculated the five trillionth binary digit of pi. (Pi is the ratio of a circle's circumference to its diameter, and its digits go on forever. Binary numbers are defined in Chapter 22's "Further Exploring" section.) In 1998, researchers computed the first 51.5 billion decimal digits of pi. Percival (shown here) discovered that pi's five trillionth bit, or binary digit, is a "0." His accomplishment is significant not only because it was a record-breaker, but because, for the first time ever, the calculations were distributed among 25 computers around the world. In all, the project, dubbed PiHex, took five months of real time to complete and one-and-a-half years of computer time. Percival, who graduated from high school in June, 1998 had been attending Simon Fraser University in Canada concurrently since he was 13.
In 1998, self-taught inventor Harlan Brothers and meteorologist John Knox developed an improved way of calculating a fundamental constant e (often rounded to 2.718). Studies of exponential growth — from bacterial colonies to interest rates — rely on e which can't be expressed as a fraction and can only be approximated using computers. Knox comments, "What we've done is bring mathematics back to the people" by demonstrating that amateurs can find more accurate ways of calculating fundamental mathematical constants. (Incidentally, e is known to more than 50 million decimal places.)

Hundreds of years ago, most mathematical discoveries were made by lawyers, military officers, secretaries, and other "amateurs" with an interest in mathematics. After all, back then, very few people could make a living doing pure mathematics. Modern-day French mathematician Olivier Gerard wrote to Dr. Googol:

This is not to say that amateurs can make progress in the most difficult-to-understand areas in mathematics. Consider, for example, the strange list in Chapter 43 that includes the ten most difficult-to-understand areas of mathematics, as voted on by mathematicians. It would be nearly impossible for most people on Earth to understand these areas let alone make contributions. Nevertheless, the mathematical ocean is wide and accommodating to new swimmers. Wonderful mathematical patterns, from intricately-detailed fractals to visually-pleasing tilings, are ripe for study by beginners. In fact, the late-1970s discovery of the Mandelbrot Set -- an intricate mathematical shape that the Guinness Book of World Records called "the most complicated object in mathematics" -- could have been made and graphically rendered by anyone with a high-school math education (see Figure here). In cases such as this, the computer is a magnificent tool that allows amateurs to make new discoveries that border between art and science. Of course, the high-schooler may not understand why the Mandlebrot Set is so complicated or why it is mathematically significant. A fully-informed interpretation of these discoveries may require a trained mind; however, exciting exploration is often possible without erudition.

End Chapter 1, Wonders of Numbers, by Cliff Pickover

Terms Indexed in Wonders of Numbers

(This is meant to give a clearer indication of topics covered.)

abcdefghij problem, 204, 360 Abundant numbers, 366 Ackermann's function, 292 Agnesi, Maria, 69, 72 Aleph naught, 91 Algebraic topology, 99 Alhazen, 310 Alien snow, 124 Aliens, 29, 60-63, 198 Amateurs, 2-6 Amicable numbers, 212-215, 362, 365 Ana sequence, 168 Ants, 15, 287 Apocalyptic numbers, 176-177, 339 Apocalyptic powers, 195, 350 Archery, 275-276 Arnold, Vladimir, 88 Arranging objects, 335 Arrow, Cupid's, 22 Artin conjecture,77, 311 Ashbacher, Charles, 97, 340 Audioactive decay, 134-138, 323 Banach Spaces, 99 Bari, Nina, 72-73 Bastions, 28 Batrachions, 220, 368 Beal, Andrew, 3 Becker, Craig, 379 Bible code, 12, 286 Big numbers, 20-21, 289-294 Binary numbers, 172 Birch and Swinnerton-Dyer Conjecture, 77 Blood, 13, 287 Borcherds, Richard, 88 Brahmagupta numbers, 180-182, 341 Bree, David, 4 Brick problem 371 Brothers, Harlan, 4 Cake cutting, 158-161, 265-266 Cakemorphic numbers, 158-161 Cantor set, 169 Cantor, Georg, 82, 310 Cardano, Gerolamo, 82 Cardinals, 99 Cards, 11, 217, 286 Carpet, Khosrow, 119, 314 Cartwright, Mary, 73 Catalan numbers, 333-335 Catalan's constant, 104 Caverns, 229-230 Cellular automata, 124-130 Chaitin's constant, 90, 104 Champernowne's number, 105 Chang, Sun-Yung Alice, 73 Cheerios, 201 Chemistry, xii Chinchillas, 257 Clarke, Arthur, 354 Clarkson, Roland, 3 Clay Mathematics Institute, 77 Codes, 267 Cohomology theory, 99 Complex numbers, 243-245 Continued fractions, 247 Conway, John, 87, 135, 323 Coprime, 317, 361 Coxeter, Harold, 85 Creator numbers, 187-188, 345 Cycles, 52-53, 307 de Fermat, Pierre, 3 Decillion, 201 Descartes, Rene, 81 Discriminant, 313 Dissection, 260-261, 271-272 Doughnut loops, 224 Dudley triangle, 144 e, 4, 68, 89, 104-106, 309 Eggs, 239-240 Einstein, Albert, vi, xii, 66 Emerald gambit, 259-260 Emirps, 178, 340 Emordnilap numbers, 142-144 Erdos, Paul xii, 63, 109-113 Euclid, 79 Euler, Leonhard, 79 Euler's constant, 90, 104 Factorial, Arabian, 183 Factorions, 358 Feigenbaum numbers, 105 Fermat, see de Fermat Fermat's Last Theorem, 3, 85 Fibonacci numbers, 173-175, 177, 219, 337-338, 354 Five, 27-32 Formulas, influential, 93-96 Forts, 28 Fractal Fractals, 5, 131-132, 169, 217-218, 222, 316, 320-322, 336 antennas, 321 checkers, 222 game, 38-39 Internet traffic, 321 number sequences, 217-218, 378 Pollock art, 321 practical, 320-322 Frame, Michael, 322 Freedman, Michael, 87 Frogs, 217, 368 Galois, Evariste, 80-81 Games, dream-worm, 50-51 fractal, 38-39 Gamma, 90, see Euler's constant Gardner, Martin, 12, 87-88 Gauss, Johann, 79 Gaussian rationals, 243-245 GCD, 315-317 Gears, 277 Germain, Sophie, 71 God, proof of, 82-83 Godel, Kurt, 82-83, 310 Goldbach Conjecture, 75 Golden ratio, 106 Goliath number, 351 Googol, Francis, viii Googol (number), 290 Gordon, Dennis, 313 Gowers, William, 88 Graham's number, 293 Grenade stacking, 206 Grothendiek, Alexander, 87 Groups, 99 Gyration, 318 Hawking, Stephen, 66-67 Hexagonal numbers, 152-154 Hexamorphic numbers, 154,159 Hexgons, 166 Hilbert, David, 77, 79 Hilbert's number, 105 Hodge Conjecture, 77 Hopper, Grace, 73 Huascaran box, 252 Hypatia, 58-60, 69 Hyperspace prisons, 147-149 Iccanobif numbers, 340 Imaginary numbers, 89, 105 IQ-Block, 42-43 Irrational numbers, 90, 367 Kaczynski, Theodore, 65, 91-92 Katydid sequences, 164, 333 Klingon paths, 46-47 Knots, 266 Knox, John, 4 Kovalevskaya, Sofia, 69-71 Kurchan array, 237 La Pileta, 9 Langlands Philosophy, 76, 311-312 Langlands, Robert, 86-87 Langlands' Functoriality, 99 Large numbers, 20-21, 289-294 Laser battle, 258-259 Latin squares, 297 Leviathan number, 196, 352 Lienhard, John, 28 Likeness sequence, 323 Liouville's number, 104 Lobsterman, 254 Looping, see Recursion Lovelace, Ada, 309 M-theory, 101 Madonna sequence, 194-195, 350 Magic squares, 4, 233-244 Magic tesseracts, 238 Mandelbrot set, 5 Mansfield, Brian, vii Mathematicians, females 69-73 influential, 78-83, 84-87 scandals, 73-75 strange, 63-66, 309-310 Mathematics, difficult, 98-102 Mazes, 55-56, 114, 248-249 McLean, Jim, 373 McMullen, Curtis, 88 Monster simple group, 293 Morse-Thue sequence, 35-36, 105, 299-300, 302 Moser, 293 Motivic cohomology , 99 Mozart numbers, 146 Mulcrone formulation, 199 Murder, 58-56 Napier, John, 82 Narcissistic numbers, 204, 358, 359 Nash, John, 65-66 Navier-Stokes existence, 77 Newton, Isaac, 78 Nicaragua, 93-96 Noether, Emmy, 71 Non-abelian reciprocity, 99 Number cave, 9 Number sequences, 44-45, 47, 194 fractal, 217-218, 366-367 Number theory, xi - xii, 111 Numbers, abundant, 366 amicable, 212-215,362, 365 apocalyptic, 176-177 binary, 172 Brahmagupta, 180-182, 341 cakemorphic, 158-161 Catalan, 333-335 complex, 243-245 creator, 187-188, 345 Emirp, 178, 340 Fibonacci, 173-175, 177, 219, 337-338, 354 Goliath, 351 gyrating, 202, 319 hailstone, 116-119, 314 hexagonaal, 152-154 Iccanobif, 340 imaginary, 89 interesting, 88-93 irrational, 90 large, 20-21, 289-294 Leviathan, 196, 352 Mozart, 146 narcissistic, 204, 359 parasite, 193-194, 348-349 perfect, 212-215, 362-366 prime, 3, 76, 138-139, 76, 92, 216, 324, 340, 342, 361-362, see also Coprime Princeton, 189-191, 348 Safford, 197 schizophrenic, 210 Smith, 247 smooth, 316 square-free, 317 sublime, 212-215, 362 symbols, 10 tower, 290 transcendental, 103-105 triangular, 149-151, 330, Ulam, 185-186 undulating, 202, 318, 355-358 untouchable, 28 vampire, 49, 306 X-Files Obsessive-compulsive disorder, 106-108 Ollerenshaw, Dame Kathleen, 4 Omega prism, 121 Ornaments, 31 Ouroborous, 47 P=NP, 76 Palindromes, 140-144, 326 Parasite numbers, 193-194, 348-349 Pascal, Blaise, 82, Pascal's triangle, 130-132, 144 Penrose, Roger, 85 Pentagonal shapes pie, 165-167 symmetry, 27-32, 165-167 Percival, 3-4 Perfect numbers, 76, 212-215, 362-366 Persistence, 54-55, 308 Peterson, Ivars, vi Pi, 3,68, 89, 104-106, 92, 309, 317, 361 Pipes, 34 Pizza cuts, 260-261 Platonic solids, 28 Poincare Conjecture, 75 Poincare, Jules, 80 Pollock, Jackson, 321 Poseidon arrays, 23 Praying triangles, 371 Primes, 3, 76, 138-139, 76, 92, 216, 324, 340, 342, 356,361-362 see also Coprime cycles, 216 records, 324, 343 Princeton numbers, 189-191, 348 Prism, omega, 121 Problems, mathematical, 74-77 Program codes, xiii Pyramids, 206 Pythagoras, x, 65 Pythagorean triangles, 27, 226-228, 370-371 Quantum groups , 99 Questions, math, 112 Quincunx, 27-32 Raedschelders, Peter, 31-32, 170-171 Raisowa, Helena, 72 Ramanujan, Srinivasa, 64-65, 66-67 Recursion, see Looping Replicating Fibonacii digits, 174-175, 337 Rice, Marjorie, 3 Riemann Hypothesis, 75 Riemann, Georg, 80 Robinson, Julia, 73 Roman numerals, 6-10, 284 RSA code, 325 Safford number, 197, 354 Scales, 24-26 Scandals, mathematical, 73-75 Scheherazade, 183 Schizophrenic numbers, 210 Schroeder, Manfred, xi, 220 Self-organizing systems, 231 Self-similarity, 36, 120, 131-132, 168, 219, 336, 367 Sequences, 44-45, 164, 194, 333 katydid, 164, 333 signature, 367 Serre, Jean-Pierre, 88 Shirriff, Ken, 344 Sierpinski triangle, 173, 130-132 Signature sequences, 367 Skewes's number, 290 Smale, Stephen, 86 Smith numbers, 247 Smith, Harry, 348 Smooth numbers, 316 Sphere packing, 243-245 Spider math, 16, 19, 288, 289 Square-free numbers, 317, 361 Squares, Latin, 297 magic, 4, 233-244, 297 mystery, 26 religious, 261 Squarions, 161-163, 332 Star Trek, 18,46 Stirling's formula, 352 String theory, 86, 99, 100 Sublime numbers, 212-215, 362 Sullivan, Michelle, 29 Symmetry, five-fold, 27-32, 165-167 Tablets, 255 Thurston, William, 86 Tiles, 3 Title, strange math, 101-103 Tower, number, 290 Transcendentals, 103-105, 211 Treadmills, 277 Trees, trivalent, 334 Triangles counting, 147-149 cycle, 19 Dudley, 144 Pascal's, 130-132, 144 Pythagorean, 27, 226-228, 370-371 Sierpinski, 173,130-132 praying, 371 Triangular numbers, 149-151, 330, Triangulation, 165-167, 147-149 Turing machine, 77 Turing, Alan, 310 Uhlenbeck, Karen, 73 Ulam numbers, 185-186, 344 Unabomber, see Kaczynski Undulation, 123, 202, 318, 355-358 Unsolved problems, 74-77 Untouchable numbers, 28 Vampire numbers, 49, 306 Wiles, Andrew, 84 Winarski, Dan, 379 Witten, Edward, 86 X-Files number, 154 Yang-Mills existence, 77 Zeta function, 105 Zoo, 253, 262 153, 359 3, 272-273 33333331, 342 365,365,365,365,365, 365 450-pound problem, 207-208 666, 176-177, 339 73,939,133, 184, 342

Reader Mail on Some of the Super-Difficult Problems

Technical Feedback

Young readers: Do not fear. There are lots of problems in the book that can be solved with pencil and paper. Here we focus on the most technical problems.

Chapters 57 and 58, Saippuakauppias and Emordnilap Numbers

From: "jar_czar" Darrell Plank:

I just got your "Wonder of Numbers" book and like it a *LOT*. In particular, the palindrome chapter got me thinking. It seems that off the top of my head, the palindrome number should grow about a digit every other step, probabilistically speaking. Since you only get palindromes only when there are no carries the real question becomes "do you ever avoid generating carries in the sequence"? Since the size of the digit sequences grows approximately linearly and the probability that carries are generated is exponentially small in the number of digits, it seems that based solely on probability, you would expect a finite probability that a particular number would never generate a palindrome. In a sense, I wonder if this sort of thing isn't a candidate for a true statement which isn't provable ala Goedel. Dunno, but it seems like the kind of case which could "happen" to be true without a proof to accompany it since you'd kind of expect it to be probabilistically true occasionally even for an infinity of numbers generated in the sequence.

Anyway, it seemed to me that it would be interesting to modify the algorithm so as to remove any powers of two from results - i.e., reverse the digits, add to the original and divide by two until you get an odd number. It seemed like this might counteract the numbers getting arbitrarily large and this "probabilistic" effect taking over. I wrote up a small Mathematica program to do this and tested it on all numbers between 1 and 10000. The most steps any number took was 27 (there are 22 of these starting at 8039 and all ending the sequence at 1148411). I took this as pretty strong evidence that the sequence was probably always finite. Interestingly enough, however, I tried a bit higher and found a handful of numbers that actually do go on infinitely, starting with 10917. They all end up in a loop which alternates between 13748625 and 16608339. I tried numbers from 10K-20K and 90K-100K and the same two step sequence was the only one that appeared. I decided to try 17000000 to 1701000 to go "beyond" the sequence and hopefully find a new sequence. Interestingly, the vast majority of cycles went back to the (13748625,16608339) cycle which I had seen earlier. One interesting variant was the cycle (137498625,166098339). In fact, it's pretty easy to see that you can put as many 9's in the middle of 1374-8625 as you like and end up with a two step cycle. You could probably come up with similar tricks to produce other cycles but I haven't thought about it much. The only other cycle that appeared was (9551509,4650767,6408413,9556459). The main interesting thing about this cycle is that it's smaller than the values in the previous cycle so obviously smaller numbers doesn't necessarily imply smaller values in the cycles.

Well, that's about all I've done tonight but there are some interesting questions to ponder (interesting in my mind anyway) - What's the smallest number which participates in a cycle? Obviously, 10917 is the smallest that ends in a cycle but it doesn't participate. Also, the larger values ended up with cycles ending up with smaller numbers so no telling if there might be smaller ones than 4650767 of the above cycle.

Are there arbitrarily long cycles? My instincts say yes, but maybe not.

Most interestingly, to me, is the question of whether any number produces an infinite sequence of non-repeating numbers. I assume not but have no proof.

Anyway, it's an interesting problem and if you hear anything more about it or any variations I'd love to know.

Once again, a *GREAT* book with a lot of other interesting questions I'll have to think about.

Thanks for the encouragement. I'll definitely look into it some more and let you know what I find. I started thinking last night about doing this in binary numbers so that the "casting out twos" is more obvious and realized that at each step you lose one or more bits and gain at most one bit so it's obvious that in binary numbers, at least, the best you can do is end up with a sequence of numbers all of which have the same number of bits (i.e., you can never generate an infinite set of numbers). Whether there are any such sequences I'm not sure. I have to get to work so don't have time to think over whether this might say something in the decimal case also.

The main reason for this mail is to correct an embarrassing typo in my last mail - the extra sequence I found has one more number in it than I wrote down - it's

(4650767,12321331,6408413,9556459,9551509).

Sorry 'bout that!

Chapter 78. Creator Numbers

Started looking over the section on minimal construction of numbers from 2 and 1 (creator numbers) and wrote a program to attempt to calculate this. The program rests on the fact that for every n, Creator(n) = Creator(i)+Creator(j) for some i,j which combine via either addition, multiplication, subtraction or powers to yield n. For every number in my range, I produced all the ways it could be a product of two numbers, all the ways it could be represented as a power and all the ways it could be represented as a sum of two numbers (I'll get to subtraction shortly). For each of these cases, the numbers combining to produce the number in question is smaller than that number so I've already calculated values for them, making it simple to calculate values for the number in question with a simple addition. I then select the operation which yields the smallest number of components and record that sum in the value array for the current number. The main fly in the ointment here is subtraction since it can produce the number in question from larger numbers. There are really two problems here - first, I don't have values for the numbers I haven't analyzed yet and second, there are an infinite number of pairs of positive integers whose difference is the number under study. For the first problem I initially set the value for all numbers to infinity. This means that no subtraction will ever be chosen because it will yield an infinite sum. To counteract this, I run through the array in a series of passes. I have to do all the operations in each pass because a smaller value due to a subtraction may allow me to multiply by that number to give a smaller number for another value. For the second problem, I only allowed numbers within the range of values I was analyzing. This means that there might be some difference of large numbers which don't lie on the table which would potentially yield a smaller representation for a number in the table. This probably doesn't affect too many numbers except a few at the end of the range being analyzed and then not by all that much typically. I also keep track of the best operation and the two values to be operated on so you can actually reconstruct the representation.

I found that while there were still differences between passes 2 and 3 there were none between passes 3 and 4 when I calculated out to 1000. 567 comes in at 8 operands using the representation (2¬(2+1)-1)*((2+1)¬2)¬2.. Likewise, 120 came in at 6 and 20 at 5 which agree with your values. I think that perhaps a difficult one might be 7 operands to get to 428.

Here are the results. They're in Mathematica format so a bit difficult to read, but you ought to be able to figure it out. The first column is the number in quotes, the second is a letter inside a bunch of formatting. The letter is either B (only for 1 and 2 - base numbers), M (for multiply), P (for power), D (for difference/subtraction) and S (for sum). This, of course, is the final operation to produce the number. The next column has two comma separated numbers which represent the operands of the operation from the second column. the last column is the minimal number of 1s and 2s found for the number. So, for instance, looking at the 7th entry you see that it's the difference between 8 and 1: 8-1. Looking at 8 you see it's the third power of 2 (2¬3-1). Looking at three, you see it's the sum of 2 and 1 (2¬(2+1)-1). There are the four operands that 7 says it requires. It should be easy to modify this to use other combinations of numbers as bases so I might play around with that in the near future.

I'm including a plot of the "creator number" for n (modulo the potential errors the subtraction may cause at the end of the range as discussed above). I hope this is the email account you can view them on.

Thanks a lot for your time!

Darrell Plank

Chapter 55, Audioactive Decay

Darrell Plank: I went ahead and wrote a mathematica program to do the "generalized" audioactive decay sequences. The mathematica program took about 10 minutes. Here it is:

nextSeq[seq_List, n_Integer] := Module[ {seqPart = Partition[seq, n, n, {1, 1}, 0]}, Flatten[Transpose[{Table[1, {Length[seqPart]}], seqPart}] //. {start___, {r_, x__}, {s_, x__}, end___} -> {start, {r + s, x}, end}]] Anyway, it seems like an interesting sequence, at least in the sense of seeming initial incomprehensibility. The 30th step for blocks of 7 is:

{1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2,
\                                                                          
1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 
1, \                                                                       
1, 0, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 2, 1, 1, 2, 2, 
1, \                                                                       
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 
1, \                                                                       
2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 
1, \                                                                       
1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 
0, \                                                                       
1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 2, 
1, \                                                                       
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 
0, \                                                                         
2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1,   
1, \                                                                         
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1,   
1, \                                                                         
1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1,   
1, \                                                                         
1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1,   
1, \                                                                         
1, 1, 1, 1, 1, 1, 2, 1, 0, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 0,   
1, \                                                                         
1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 0, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1,   
1, \                                                                         
1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,   
0, \                                                                         
1, 0, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0,   
1, \                                                                         
0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1,   
1, \                                                                         
0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0,   
1, \                                                                         
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0,   
1, \                                                                         
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 2, 1, 1, 1, 1, 1, 1,   
1, \                                                                         
1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1,   
1, \                                                                         
2, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0,   
1, \                                                                         
1, 2, 1, 0, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 1,   
0, \                                                                         
1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 1,   
0, \                                                                         
0, 1, 0, 0, 0, 0, 0, 0, 0}

As you can see, no 3s appear just yet, but I checked and one 3 appears somewhere before the 50th step which has 2500 or so terms. The 0's from the end of the sequence have really wormed their way into the sequence, occurring first at element 53. I also found some 4's in the length 3 block sequence so we do get values greater than 3 which was kind of my goal all along.

If 333 appears, then at least one 3 must be a count and the following 3 be the value which implies that 333 appeare in the previous string in the sequence. Conversely, the absence of 333 in a string means no 333 in the next string. Since the first string doesn't have 333 then by induction none of them do.

Chapter 80, Parasite Numbers

Darrell Plank: Hey Cliff -

I woke up in the middle of the night and started thinking about your "parasite" numbers. Let's take your first example: 4 * 102564 = 410256. Call the multiplier m (4 in this case), the one with the digit on the right R (102564) and the one with the digit on the left L (410256) and the digit switching positions d (4). Then from the equation mR = L we have:

10L - R = 10 m R - R = R(10m - 1)

while from the definition of "parasite number" we have:

10L - R = d*10¬n - d = d(10¬n - 1)

for some n so that 10m - 1 has to divide d(10¬n - 1) for some n. This seemed easy enough to program up in Mathematica so I tried it and came up with a huge number of these things. One interesting thing is why 4 is so prevalent - mainly because 4 * 10 - 1 = 39 = 3 * 13 is easier to divide into an integer composed only of 9s (10¬n - 1) than say, 3 * 10 - 1 = 29. You got all the ones < 1,000,000 unless you count the "cheaters" which have a "0" at the front of R (025641 * 4 = 102564 for instance). At least up to the first 50 digits, the only other ones are ones which repeat the patterns in the solutions < 1 million. For instance, 102564102564 * 4 = 410256410256. Obviously, you can extend these to any length you like.

You have to go further with other digits, but the parasites are out there. You pointed out one with 5: 142857. Similarly to the case for 4, this one repeats for a very long time until suddenly we get another very big number:

5 * 020408163265306122448979591836734693877551 = 102040816326530612244897959183673469387755

which is a cheater. The first non-cheater which isn't an extension of the original is a true parasite per your definition rather than a psuedo-parasite:

5 * 10204081632653061224489795918367346938775 = 51020408163265306122448979591836734693877

which is obviously a cyclic permutation of the cheater above. All of the others under 50 digits seem to be cyclic permutations of this number or repeats of the first number.

3 produces 3 * 344827586206896551724137931 = 134482758620689655172413793 in various cycles.

A true parasite for 2 is 2 * 105263157894736842 = 210526315789473684. As usual, various cycles.

6 shows nothing until 60 digits or so which I'm not even going to attempt to write down (although I can send all these results to you if you really want to look at them).

7 has the more reasonable 7 * 1014492753623188405797 = 7101449275362318840579. Actually 7 seems kinda interesting. This is just the first true parasite. There are several smaller parasites. In fact, there's a 22 digit psuedo-parasite which switches every digit from 1 to 9. I think they're all multiples of 144927536231884057971. Further numbers are just repeats of this pattern.

8 displays similar characteristics, all the numbers being multiples of 126582278481. Thus, 8 * 126582278481 = 1012658227848 (cheater) and multiples of this up through 9.

9 is similar, starting with the largish cheater 9 * 11235955056179775280898876640449438202247191 = 101123595505617977528089887664044943820224719

Whew! So much for the short enumeration. A lot of interesting questions left over - how about switching n digits from the front to the back? The same method I used for these ought to work for n digits I think. Similarly, numbers other than single digits ought to be able to be checked. Just had a thought - it would be interesting if you could find two numbers whose multiple also factored into two other numbers which were cyclic permutations of the original factors. Wonder if that's possible?

Also, there's a lot of interesting stuff in the patterns found in the cases I enumerated. All single digits except 1 and 0 (obviously) have at least psuedo primes of one digit. Is it true for all numbers in general? Do the numbers break out of the patterns obvious in their parasites less than 50 digits? the fact that 5 goes a long ways with one pattern and then breaks into another gives me hope. Also, the rather chaotic nature of which digits composed solely of 9s are divisible by 10m-1 makes me rather hopeful. I'm not sure how much can be proven.

Sorry for the long mail but I thought it all turned out kind of interesting.

Darrell

Chapter 55

From [email protected]

First off, let me say that your most recent book is wonderfully thought provoking as always. I am glad to have discovered your writings and thank you for doing your part to make the world such a fascinating place (or if you prefer, for revealing the underlying fascinations that so few people take the time to see).

I am writing to make a comment about one chapter in your most recent book, "Wonders of Numbers." Pleas correct me if I am wrong (and if I am not, then there have surely been hundreds here before me), but in Chapter 55, where you discuss the likeness sequence, you ask whether it can be proven that "3-3-3" cannot occur. It seems to me that it cannot occur for the following reason (please excuse the lack of mathematical rigor):

Given the structure of the sequence, in any threesome of numbers there will be a pair whose meaning is "n" occurrences of the number "m". this means that the sequence 3-3-3 must contain a pair whose meaning is "3 occurrences of the number 3". But this is recursive-you cannot get a 3-3-3 without already having had an instance of the same. So given that the sequence does not include the sequence 3-3-3 at the start, it will never occur spontaneously.

Regards . . . Brian

Hexbonacci Numbers

From: Sarn Ursell

dearest Clifford A Pickover,

I have veiwed you"re website with great interest, and indeed, I would be simply honoured for you to allow me to include a link to you"re web pages.

I saw you're latest book WONDERS OF NUMBERS, and indeed, I am ITCHING to crack into this text.

The main point of this email is to suggest to you a set of number sereis that you may even consider writeing about in a later book, and these sereis is indeed, something I will be placeing into my site.

You will no doubt know about the Fibonacci sereis, which is simply made by starting with two ones and adding terms one behind to make the new:

T_n+1=T_n+T_n-1

However, you may have heard about the Tribonacci series, which is made by adding the last two digits:

1, 1, 2, 4, 7, 13, 24, 44, 81s.

And from this the quadbonacci series, the pentbonacci sereis, and the hexbonacci series, all the way up to the n-bonacci sereis.

Each ratio of sucessive terms forms a special constant.

Food for thought, so please do check them!

Your friend,

Sarn.

Chapter 31 "A Ranking of the 5 Strangest Mathematicians Who Ever Lived

From: "Jason"

Dr. Pickover,

I have been enjoying your book "Wonders of Numbers" lately and noticed that you included John Nash in Chapter 31 "A Ranking of the 5 Strangest Mathematicians Who Ever Lived." Maybe you are already aware of it but he has a "web page" at Princeton: www.math.princeton.edu/~jfnj/ that you could have directed your readers to. He has many texts available on-line that are quite intriguing. I especially like the "HILdos38.txt" file under the logic section which has some new ideas on the extension of logical systems. He also has listings of various Mathematica programs he wrote. One related to the Goldbach conjecture. I noticed you did have a link on your homepage to a review of Nassar's book on him. But not to his homepage.

Not too long ago I sent a "fan email" to Dr. Nash and he responded. I was thrilled! I just thought you might like to know about his home page in case any other of your readers might be interested, and I am really enjoying "Wonders of Numbers."

Jason

Chapter 10. Numbers Beyond Imagination

Dear Clifford A Pickover,

I am pleased to say that I finally veiwed your book THE WONDERS OF NUMBERS, and I liked the concepts contained within.

One of the quirky little exercises that you included was the creation of the biggest number possible from the digits: 1, 2, 3, 4, 5, two parenthesis, one minus sign, and a full stop.

I was quite taken aback, but I had a funny little idea when walking home from the bookstore in my home country Wellington, New Zealand.

I want to introduce you to a form of specialized notation called "Arrow notation", and it is from this that a VAST range of deep, and mysterious experiments can occur.

A+b=a[1]b, a*b=a[2]b, a¬b=a[3]b, and so on.

The use of the number [4] in brackets is actually what is called a "superpower", aka a tower exponent, and I am actually quite bemused that you haven't said anything about this operator in any of your books.

I regretfully, cannot include any form of this arrow notation, or use any specialized glyphs to send this data to you, but you can, (and I feel you will), use you"re imagination.

Imagine the numbers in these square brackets placed on top of an upward pointing arrow.

It is from this that any number can be placed on top of this.

What I am looking for, however, is a formal definition of non-integer, negative, imaginary, complex, and hyper-complex up arrows, and these as applied to higher powers, as in the place of c in a[4, 5, 6, 7, 8,s..n]c.

This is all easier said than done, and I am undertaking a formal course in mathematics to awnser some of these questions.

If you can imagine a triangle system containing these numbers:

                                                                       
                                                                         
       /\                                                                
      /4 \                                                               
     /----\          =   65536                                           
    /2 | 4 \                                                             
   ---------

This is, by the way, merely another variant on this generalized arrow notation, but the funny thing is about this is that we can form some VERY oddball geometry from this.

If you can imagine joining up the base in the shape of a triangle to the apex of this as a prymid, as in achinent Egypt, then you"re on the right track.

I could have thought that the VOLUME of this shape could be equal to 65536 units, and the height of the triangle equal to 4 units, and the sides 2 units left from the center, and 4 units right from the center.

Now, this is where it gets really interesting, and this is where by we "splice in" non-integer up arrows between 4 and 4 to equal 65536, and between 2 and 4 to equal 65536.

I know that 4[3]4=256, and that 2{4]4=65536.

This would mean splicing in "in-between points" which were four units high, and 3.k units high between the four up arrow and the 4 to the right.

Prime Numbers

From: "Pedro Caceres"

Hi,

my name is Pedro and I have just read half of your book "Wonders of Numbers". It is really good. Anyway, I want to share a curiosity with someone that is living in the mathematical world.

I was playing around the prime numbers and looking for any kind of relationship among them and I came up with the idea that they may be a combination of two different series. I worked this idea out and I realized that I could generate the prime series with the following two formulas :

                                                                 
        Aseries =3D (6n+1)  with n=3D0,1,2...    resulting : 1,7,13,19,    
        Bseries =3D (6n+5)  with n=3D0,1,2...    resulting : 5,11,17,23,

I could not generate the prime numbers 2,3 (maybe because they generate the base number 6!).

If I put the two series together I come up with :

1, (2,3), 5,7,11,13,17,19, 23 ...

if we continue both series non-prime numbers appear "contaminating" the prime series,

1, (2,3), 5,7,11,13,17,19, 23, 25, 29, 31, 35, 37, 41, 43, 47, 49, 53

but these non-primes numbers are the multiplication of the previous prime numbers (not just the odd numbers), that we will write as :

mul(5) =3D 5*5, 5*7, 5*11, 5*13, ...
mul(7) =3D 7*7, 7*11, 7*13, 7*17, ...

all these prime factors have already been generated previously so we can use them to check the prime condition of any new number.

In essence, the A series & B series combined can be written as : 1, (2,3), mul(5), mul(7), mul(11), mul(13), mul(17), mul(19), mul(23), ...

At least I can say that any prime number will be a member of the A or/and B series.

Just a curiosity, is this interesting at all?

Thanks Pedro Caceres

Chapter 65, Cakemorphic Numbers

From: "Darrell Plank"

I was looking over the book this afternoon and noticed the devilishly fascinating cakemorphic integers - number such that (n¬2 + n + 2)/2 ends with the same digits. In general, the units digits repeat themselves every 10 integers for any polynomial with integral coefficients. Same applies to last two digits every 100 integers, etc.. Why? Because all the operations in an integral coefficient polynomial preserve remainders.

So in the case of n¬2 + n + 2 we've got (mod 10).

Now when you divide, modular arithmetic states that if a=3Db(mod c) and d =3D GCD(a,b,c) then a/d =3D b/d (mod c/d). So when you divide by two in the cake formula we have to not only divide the above remainders by 2 but we have to take them mod 5 which is the same as saying that they're either the original number/2 or that number plus 5. Thus (again, mod 10)

                                                                       
n             (n¬2 + n + 2) / 2                                          
0             1,6                                                        
1             2,7                                                        
2             4,9                                                        
3             2,7                                                        
4             1,6                                                        
5             1,6                                                        
6             2,7                                                        
7             4,9                                                        
8             2,7                                                        
9             1,6

As you can see, none of the digits on the right is the same as the digit on the left above, which translates to no integer ever has the same ones digit after running it through the cake formula which of course means that none of them have then same number in the last n digits.

Darrell

More research and software relating to Wonders of Numbers is presented at this web page.