"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."
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...

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

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

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

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

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,

- 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,

- 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,

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*. The six letters represent integers, with^{z}*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 3^{6}+ 18^{3}= 3^{8}, 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:

- I believe that amateurs will continue to make contributions to science and mathematics. Computers and networks allow amateurs to work as efficiently as professionals and to cooperate with one another. When one considers the time wasted by many professionals in grant writing and for other paperwork justifying their activity, the amateurs may even have a slight edge in certain cases. However, the amateurs often lack the valuable experience of teaching or having a mentor.

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

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.

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!

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

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.

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

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

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.

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

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.

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

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 1,6As 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

John Vickers writes:

Dear Cliff,

You ask in "Wonders of Numbers" whether there are any patterns in the digits of the Robins Princeton Numbers given by R(n)= Product of (3i+1)!/(n+i)! where i ranges from 0 to n-1. In particular you ask about the number of trailing zeroes. I enclose the number of trailing zeroes for the first 10000 Robins Numbers. I could enclose my code if you are interested. A zero will only occur at the end of a base 10 number if you have multiplied somewhere by 10=2*5. But to count the number of 5s or 2s occuring in any factorial x! you just add [x/5]+[x/5*5]+[x/5*5*5] +.... where [] means the integer part. So I add the number of fives in the top collection of factorials together and subtract all the ones coming from the denominator, do the same for the twos and the number of zeroes in the Robbins Princeton Number is the minima of the two numbers. I dont have to work out a single factorial to do this. The first 25 numbers in my file of 10000 which I calculated last night agrees with the R(n) given in the book "Wonders Of Numbers". This excludes a large number of possible numbers being "Morphic" as you asked or having themselves as there own trailing digits. If you notice in the file there are long segments of Robins numbers which increase in the number of trailing zeroes by 1 and then decrease again just like your computed first 25. John Vickers

Enclosed is a simple proof that certain Katydid Sequences do not contain repeats, I do it for the cited 2x+2,6x+6 sequences in your "Wonders Of Numbers"

There are no repeats in the sequences x-à2x+2, x-à6y+6.

Proof)By looking at where repeat first occurs and tracing backwards to find contradiction. If there was a repeat, let (x,y) be first pair Then x,y>1 X=y If x comes by applying xà2x+2 then y comes from applying yà6y+6 Otherwise (x,y) wouldn't be minimal So 2x+2=6y+6.; If x=1 now then lhs=4 #; If y=1 now then rhs=12 so x=5 # because all numbers occurring on all branches are even apart from 1; So now x,y>1

Case 1; X=2x+2; Then2(2x+2)+2=6y+6; 4x+6=6y+6; 4x=6y; 2x=3y; x=1 # y=1#; Case 1a; Y=2y+2; 2x=3(2y+2)=6y+6; x=3y+3; y=1 x=6 doesn't ccur#; So y is even So 3y is even So 3y+3 is odd; So x is odd impossible #;

Case 1a; Y=6y+6; 2x=3(6y+6)=18y+18; y=1 -à x=18 doesn't occur#; x=9y+9; y even à 9y even à9y+9 odd; x odd #;

Case 2 x=6x+6; 2(6x+6)+2=6y+6; 12x+12+2=6y+6; 12x +14=6y+6; 6(y-2x)=14-6=8; lhs=0 mod(6) rhs=2 mod(6) #;

QED Similar no Katykid repeat for x->2x+2, y->4y+4

page 205, boxed equation, Chapter 89, near bottom: "c" should be superscripted

page 358, Chapter 89, 2nd line in Further Exploring: "c" should be superscripted

page 216, Chapter 95, line 3, missing upside-down F symbol. Should read "F(24)" not "(24)", where the "F" is the upside-down symbol. Similarly line 2 should read "Let's define a new function F(n)."

page 262, figure drawing error. Line segment misplaced. Figure 114.1 should have a line connecting point B to the dot directly to the the left instead of the line from the point below B to the point directly to the left.

page 94, Equation 10 towards the bottom of the page is missing an italics "i" before the "sin".

See Pickover books at Amazon.Com in separate window.

Return to Cliff Pickover's home page which includes computer art, educational puzzles, higher dimensions, fractals, virtual caverns, JAVA/VRML, alien creatures, black hole artwork, and animations.