END NOTES INTRODUCTION IE.T. Bell, Men of Mathematics (New York: Simon & Schuster, 1937), p. 405. 2Stephen Hawking, ed.,A Brief History of Time: A Reader's Companion (New York: Bantam Books, 1992), p. vii. 3Intemet: http://www.groups.dcs.st-and.ac.uk:80/ -history I 4Intemet: [email protected]5Intemet: wuarchive.wustl.edul docl misc/pi CHAPTER 1 IMark Twain, The Adventures of Huckleberry Finn (Franklin Center, PA: The Franklin Library, 1983), p. 19. 2Donald R. Griffin, Animal Thinking (Cambridge, MA: Harvard University Press, 1984); Guy Woodruff and David Pemack, "Primate Mathemati- cal Concepts in the Chimpanzee: Proportionality and Numerosity," Nature, Vol. 293, October 15, 1981, 568. 3Recent evidence suggests that Homo erectus may be much older-as much as 2.5 million years old. 4Denise Schrnandt-Besserat, Before Writing (Austin, TX: University of Texas Press, 1992). 5Jacques Soustelle, Mexico (New York: World Publishing Company, 1967), p.125. 6Peano's axioms actually contain the primitive term "zero" rather than "1." For this illustration I have begun the number sequence with 1 rather than zero. 7Kathleen Freeman, Ancilla to the Pre-Socratic Philosophers (Cambridge, MA: Harvard University Press, 1966), p. 75. BSir Thomas Heath, A History of Greek Mathematics (London: Oxford Uni- versity Press, 1921), p. 75. 297
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END NOTES
INTRODUCTION IE.T. Bell, Men of Mathematics (New York: Simon & Schuster, 1937), p. 405. 2Stephen Hawking, ed.,A Brief History of Time: A Reader's Companion (New
York: Bantam Books, 1992), p. vii. 3Intemet: http://www.groups.dcs.st-and.ac.uk:80/ -history I 4Intemet: [email protected] 5Intemet: wuarchive.wustl.edul docl misc/pi
CHAPTER 1 IMark Twain, The Adventures of Huckleberry Finn (Franklin Center, PA: The
Franklin Library, 1983), p. 19. 2Donald R. Griffin, Animal Thinking (Cambridge, MA: Harvard University
Press, 1984); Guy Woodruff and David Pemack, "Primate Mathematical Concepts in the Chimpanzee: Proportionality and Numerosity," Nature, Vol. 293, October 15, 1981, 568.
3Recent evidence suggests that Homo erectus may be much older-as much as 2.5 million years old.
4Denise Schrnandt-Besserat, Before Writing (Austin, TX: University of Texas Press, 1992).
5Jacques Soustelle, Mexico (New York: World Publishing Company, 1967), p.125.
6Peano's axioms actually contain the primitive term "zero" rather than "1." For this illustration I have begun the number sequence with 1 rather than zero.
7Kathleen Freeman, Ancilla to the Pre-Socratic Philosophers (Cambridge, MA: Harvard University Press, 1966), p. 75.
BSir Thomas Heath, A History of Greek Mathematics (London: Oxford University Press, 1921), p. 75.
297
298 MATHEMATICAL MYSTERIES
9 Aristotle, The Basic Works of Aristotle, trans. J. Annas (Richard McKoen, ed.) (New York: Random House, 1941); The Metaphysics, 986a, lines 15-18, Oxford University Press.
IOJames R. Newman, "The Rhind Papyrus," in The World of Mathematics, Vol. 1, ed. James R. Newman (New York: Simon and Schuster, 1956), p. 174.
IIHeath, A History of Greek Mathematics, p. 76. 12Intemet: sci. math, Alex Lopez-Ortiz, University of Waterloo, alopez
[email protected], 6/16/94. 13Ibid. 14Philip J. Davis, The Lore of Large Numbers (New York: Random House,
1961), p. 23. ISIntemet: sci.math, Lee Rudolph, Department of Mathematics, Clark
CDs, Inc., 1994). 2Lucas Bunt, Phillip Jones, and Jack Bedient, The Historical Roots of Elemen
tary Mathematics (New York: Dover Publications, 1976), p. 86. 3Florian Cajori, A History of Elementary Mathematics (New York: Macmillan
& Company, 1930), p. 44. 4David Eugene Smith, History of Mathematics (New York: Dover Publica
tions, 1951), p. 143. S N (aleph) is the first letter of the Hebrew alphabet and is commonly used
to represent infinite sets.
CHAPTERS lRalph Waldo Emerson, Representative Man (CD: DeskTop BookShop)
(Indianapolis: WeMake CDs, Inc., 1994). 2The first to prove this remarkable result was actually the Englishman,
Richard Suiseth (fl. ca. 1350), who gave a long verbal proof of an equivalent expression.
30thers were also working along the same lines at this time. In fact, Jobst Burgi (1552-1632) of Switzerland may have developed a similar system as early as 1588, but did not publish his ideas until 1620, after the work of Napier had already appeared.
CHAPTER 6 IThomas Paine, Address to the People of England (CD: DeskTop BookShop)
(Indianapolis: WeMake CDs, Inc., 1994). ~arl B. Boyer, A History of Mathematics (New York: John Wiley and Sons,
1968), p. 55. 3H.E. Huntley, The Divine Proportion (New York: Dover Publications, 1970),
p.62. 4David Wells, The Penguin Dictionary of Curious and Interesting Numbers
(New York: Penguin Books, 1986), p. 37. SBoyer, A History of Mathematics, p. 281. 6The Fibonacci Quarterly, Fibonacci Association c/o South Dakota State
University Computer Science Department, Box 2201, Brookings, SD 57007-1596.
7H.E. Huntley, The Divine Proportion, p. 160.
300 MATHEMATICAL MYSTERIES
BGeorge Cheverghese Joseph, The Crest of the Peacock (London: Penguin Books, 1991), p. 197.
9Jean L. McKechnie, ed., Webster's New Twenthieth Century Dictionary of English Language (New York: Simon and Schuster, 1979), p. 1834.
CHAPTER 7 IH.G. Wells, The Time Machine (CD: DeskTop BookShop) (Indianapolis:
WeMake CDs, Inc., 1994). 2Paulo Ribenboim, The Little Book of Big Primes (New York: Springer-Verlag,
1991), p. 142. 3For a further discussion of this equation see Paulo Ribenboim, The Book of
Prime Number Records, second edition (New York: Springer-Verlag, 1989), p. 190.
CHAPTERS IJohn Locke, An Essay Concerning Human Understanding (CD: DeskTop
BookShop) (Indianapolis: WeMake CDs, Inc., 1994). 2Paulo Ribenboim, The Book of Prime Number Records, second edition (New
York: Springer-Verlag, 1989), p. 130. 3Ibid, p. 136. 4Ibid, p. 144. 5Since Samuel Yates' death, the list of Titanic primes has been kept by
Professor Chris Caldwell who is kind enough to make the list available to interested parties. If you would like a list of the largest primes send $4 to: Professor Chris Caldwell Department of Mathematics University of Tennessee at Martin Martin, TN 38238 If you have access to the Internet, Professor Caldwell maintains a home page on the World Wide Web containing the largest primes: http://www.utm.edu:80/research/primes/largest.html. The records for largest primes of various types listed here all come from Caldwell's Internet list.
6R 1031 was discovered by Williams and Dubner in 1986. 7Ribenboim, The Book of Prime Number Records, p. 286.
CHAPTER 9 IHerodotus, The History of Herodotus (CD: DeskTop BookShop) (Indianapo
lis: WeMake CDs, Inc., 1994).
END NOTES 301
2D. James Bidzos and Burt S. Kaliski Jr., "An Overview of Cryptography," LAN TIMES, February 1990.
3Whitfield Diffie and Martin E. Hellman, "Privacy and Authentication: An Introduction to Cryptography," Proceedings of the IEEE, Vol. 67, No.3, March 1979,397-427.
4Ronald L. Rivest, Adi Shamir, and Leonard Adleman, "A Method for Obtaining Digital Signatures and Public-Key Cryptosystems," Communications of the ACM, Vol. 21, No.2, February 1978, 120-126.
5William Booth, "To Break the Unbreakable Number," The Washington Post, June 25,1990, A3.
6Paulo Ribenboim, The Book of Prime Number Records, second edition (New York: Springer-Verlag, 1989), p. 478.
7Barry A. Cipra, "PCs Factor a 'Most Wanted' Number," Science, Vol. 242, December 23,1988,1634.
8Private phone conversation with Kurt Stammberger, Technology Marketing Manager, RSA Data Security, Inc., June 22,1995.
91 must give special thanks to both Professor Ronald Rivest of MIT and Kurt R. Stammberger, Sales and Marketing Manager for RSA Data Security, Inc., for all the information they provided to me on the RSA cryptosystem and the RSA factorization contest.
lORSA Laboratories, CryptoBytes, Vol. 1, No.1, Spring 1995, l. llWant to try your hand at additional numbers in the challenge? For more
information on the contest and a copy of the numbers write to: RSA Challenge Administrator 100 Marine Parkway, Suite 500 Redwood City, CA 94065 (415) 595-8782 or send e-mail to:[email protected]. or browse the World Wide Web page, http://www.rsa.com.
12RSA Laboratories, "Answers to Frequently Asked Questions," revision 2.0, October 5, 1993.
CHAPTER 10 lHenry David Thoreau, Walden (CD: DeskTop BookShop) (Indianapolis:
WeMake CDs, Inc., 1994). 2Robert Kanigel, The Man Who Knew Infinity (New York: Charles Scribner's
Sons, 1991), p. 11. 3Ibid, p. 71. 4Ibid, p. 86. 5Bruce C. Berndt, Ramanujan's Notebooks, Vol. I (New York: Springer Verlag,
1985), p. 2. 6Ibid, p. 4. 7Robert Kanigel, The Man Who Knew Infinity, p. 203.
302 MATHEMATICAL MYSTERIES
BJames R. Newman, ed., The World of Mathematics, Vol. 1 (New York: Simon and Schuster, 1956), pp. 371-372.
9G. H. Hardy, A Course of Pure Mathematics (London: Cambridge University Press, 1963); G.H. Hardy and E.M. Wright, An Introduction to the Theory of Numbers (New York: Oxford University Press, 1979).
lOJames R. Newman, ed., The World of Mathematics, p. 2029. llRobert Kanigel, The Man Who Knew Infinity, p. 347. 12James R. Newman, ed., The World of Mathematics, p. 2038. 13Ibid, pp. 2027-2029. 14Jerry P. King, The Art of Mathematics (New York: Plenum Press, 1992), p.
29.
CHAPTER 11 IPlato, The Dialogues of Plato, trans. B. Jowett (New York: Random House,
1937), TImaeus, p. 66. 2Euclid, Elements, Book V (New York: Dover Publications, 1956), p. 139. 3The majority of the following Ramanujan equations have been taken from
Bruce C. Berndt, Ramanujan's Notebooks, Vol. I & II (New York: Springer Verlag, 1985).
4Robert Kanigel, The Man Who Knew Infinity, p. 247. 5David Wells, Curious and Interesting Numbers (London: Penguin Books,
1986), p. 100. 6Konrad Knopp, Theory and Application of Infinite Series (New York: Dover
Publications, 1990), p. 548. 7Ivars Peterson, Islands of Truth: A Mathematical Mystery Cruise (New York:
W.H. Freeman and Company; 1990), p. 177.
CHAPTER 12 IDavid Hume, An Enquiry Concerning Human Understanding (CD: DeskTop
BookShop) (Indianapolis: WeMake CDs, Inc., 1994). 2Internet: scLmath, Kent D. Boklan, July 21,1994. 3Paulo Ribenboim, The Book of Prime Number Records, p. 230. 4Internet: sci.math, Boklan, July 21,1994. 5Paulo Ribenboim, The Book of Prime Number Records, p. 230. 6Internet: sci.math, Chris Thompson, August 5,1994. 7Henry F. Fliegel and Douglas S. Robertson, "Goldbach's Comet: The
Numbers Related to Goldbach's Conjecture," Journal of Recreational Mathematics, Vol. 21, No.1, 1989, 1-7.
BG.H. Hardy and E.M. Wright, An Introduction to the Theory of Numbers (Oxford, England: Clarendon Press, 1979), p. 358.
9Robert Kanigel, The Man Who Knew Infinity, p. 135.
END NOTES 303
lomformation of g(n) taken from Paulo Ribenboim, The Book of Prime Number Records, pp. 240-245.
CHAPTER 13 1Robert Kanigel, The Man Who Knew Infinity (New York: Charles Scribner's
Sons, 1991), p. 220. 2Paulo Ribenboim, College Mathematics Journal, Vol. 25, No.4, September
1994,288. 30ystein Ore, Number Theory and Its History (New York: Dover Publica
tions, 1976), p. 78. 4E.T. Bell, Men of Mathematics, p. 486. 5E.T. Bell, Mathematics: Queen and Servant of Science (New York: McGraw
Hill Book Company, 1951), p. 194. 6c. Stanley Ogilvy and John T. Anderson, Excursions in Number Theory
(New York: Dover Publications, 1966), p. 96. 7Paulo Ribenboim, The Book of Prime Number Records (New York: Springer
Verlag, 1989), p. 180. 8Keith Devlin, Mathematics: The New Golden Age (New York: Penguin
Books, 1988), p. 210. 9D.E. Smith, History of Mathematics, Vol. 1 (New York: Dover Publications,
1951), p. 504. loDevlin, Mathematics: The New Golden Age, p. 218. llRibenboim, The Book of Prime Number Records, p. 170. 12Devlin, Mathematics: The New Golden Age, p. 219 11an Stewart, The Problems of Mathematics (New York: Oxford University
Press, 1987), p. 126. 14James R. Newman, ed., The ~orld of Mathematics (New York: Simon and
Schuster, 1956), p. 2026. 1sDevlin, Mathematics: The New Golden Age, p. 215.
CHAPTER 14 1John Locke, An Essay Concerning Human Understanding (CD: DeskTop
BookShop) (Indianapolis: WeMake CDs, Inc., 1994). 2J.M. Dubbey, Development of Modern Mathematics (New York: Crane, Rus
sak & Company, 1970), p. 112. ~udy Rucker, Infinity and the Mind (New York: Bantam Books, 1982), p.
177. 4James R. Newman, "The Foundations of Mathematics," in The World of
Mathematics (New York: Simon and Schuster, 1956), p. 1616. SDubbey, Development of Modern Mathematics, p. 113. 6Rucker, Infinity and the Mind, p. 169.
304 MATHEMATICAL MYSTERIES
7Emest Nagel and James R. Newman, Codet's Proof(New York: New York University Press, 1958), p. 3.
BMost of the following terminology and examples are taken from the Nagel and Newman book, Codet's Proof.
9Ibid, p. 68. JOIan Stewart, The Problems of Mathematics (New York: Oxford University
Press, 1987), p. 218. llJohn D. Barrow, Pi in the Sky (New York: Little, Brown and Company,
1992), p. 122. I2Rucker, Infinity and the Mind, p. 176. I3Barrow, Pi in the Sky, p. 117. I4Ibid, p. 123. I5Rucker, Infinity and the Mind, p. 181. 16Barrow, Pi in the Sky, p. 261. I7Dougias R. Hofstadter, Metamagical Themas: Questing for the Essence of
Mind and Pattern (New York: Basic Books, Inc., 1985). 1BKathleen Freemen, Ancilla to The Pre-Socratic Philosophers (Cambridge,
MA: Harvard University Press, 1966), p. 9. I9Hofstadter, Metamagical Themas, pp. 11-13. 2°Ibid, p. 14. 2INagei and Newman, Codet's Proof, p. 97. 22Thomas Paine, Age of Reason (CD: DeskTop BookShop) (Indianapolis:
WeMake CDs, Inc., 1994).
SUGGESTED READING
Aristotle, The Basic Works of Aristotle, ed. Richard McKeon (New York: Random House, 1941).
John D. Barrow, Pi in the Sky (New York: Little, Brown and Company, 1992). Albert H. Beiler, Recreations in the Theory of Numbers (New York: Dover
Publications, 1966). Bruce C. Berndt, Ramanujan's Notebooks, Vol. I & II (New York: Springer
Verlag, 1985). Eric Temple Bell, Mathematics: Queen and Servant of Science (New York:
McGraw-Hill Book Company, 1951). Eric Temple Bell, Men of Mathematics (New York: Simon & Schuster, 1965). Eric Temple Bell, The Magic of Numbers (New York: Dover Publications,
1974). E. J. Borowski and J.M. Borwein, The HarperCollins Dictionary of Mathemat
ics (New York: HarperCollins Publishers, 1991). Carl B. Boyer, A History of Mathematics (New York: John Wiley and Sons,
1968). T.J. Bromwich, An Introduction to the Theory of Infinite Series (New York:
Chelsea Publishing Company, 1991). Lucas Bunt, Phillip Jones, and Jack Bedient, The Historical Roots of Elemen
tary Mathematics (New York: Dover Publications, 1976). Florian Cajori, A History of Elementary Mathematics (London: MacMillan
Company, 1924). Thomas Crump, The Anthropology of Numbers (New York: Cambridge
University Press, 1990). Joseph Warren Dauben, Georg Cantor: His Mathematics and Philosophy of the
Infinite (Princeton, New Jersey: Princeton University Press, 1979). Donald M. Davis, The Nature and Power of Mathematics (Princeton, New
Jersey: Princeton University Press, 1993). Philip J. Davis, The Lore of Large Numbers (New York: Random House, 1961).
305
306 MATHEMATICAL MYSTERIES
Korra Deaver, The Master Numbers (Alameda, California: Hunter House, 1993).
Richard Dedekind, Essays on the Theory of Numbers (La Salle, Illinois: Open Court Publishing Company; 1948).
Keith Devlin, Mathematics: The New Golden Age (London: Penguin Books, 1988).
Heinrich Dorrie, 100 Great Problems of Elementary Mathematics (New York: Dover Publications, 1965).
J.M. Dubbey; Development of Modern Mathematics (New York: Crane, Russak & Company; 1970).
William Dunham, Journey Through Genius: The Great Theorems of Mathematics (New York: John Wiley & Sons, 1990).
Euclid, Elements (New York: Dover Publications, 1956). Graham Flegg, Numbers: Their History and Meaning (New York: Schochken
Books, 1983). Graham Flegg, Numbers Through the Ages (London: MacMillan Education
Ltd,1989). Kathleen Freeman, Ancilla to the Pre-Socratic Philosophers (Cambridge,
Massachusetts: Harvard University Press, 1966). J. Newton Friend, Numbers: Fun & Facts (New York: Charles Scribner's
Sons, 1954). J. Newton Friend, More Numbers: Fun & Facts (New York: Charles Scrib
ner's Sons, 1961). Richard Gillings, Mathematics in the Time of the Pharaohs (New York: Dover
Publications, 1972). Migene GonzeUez-Wippler, The Complete Book of Spells, Ceremonies, and
Magic (New York: Crown Publishers, 1978). Jacques Hadamard, The Psychology of Invention in the Mathematical Field
(New York: Dover Publications, 1945). G.H. Hardy; A Course of Pure Mathematics (Cambridge, England: Cam
bridge University Press, 1963). G.H. Hardy and E.M. Wright, An Introduction to the Theory of Numbers
(Oxford: Clarendon Press, 1979). Sir Thomas Heath, A History of Greek Mathematics (London: Oxford Uni
versity Press, 1921). Douglas R. Hofstadter, Metamagical Themas: Questingfor the Essence of Mind
and Pattern (New York: Basic Books, Inc, 1985). Stuart Hollingdale, Makers of Mathematics (London: Penguin Books, 1989). Ross Honsberger, Mathematical Gems II (Washington, DC: The Mathemati
cal Association of America, 1976). Ross Honsberger, Mathematical Plums (Washington, DC: The Mathematical
Association of America, 1979). H.E. Huntley; The Divine Proportion (New York: Dover Publications, 1970).
SUGGESTED READING 307
A.E. Ingham, The Distribution of Prime Numbers (Cambridge, England: Cambridge University Press, 1990).
George Cheverghese Joseph, The Crest of the Peacock: Non-European Roots of Mathematics (London: Penguin Books, 1991).
E. Kamke, Theory of Sets (New York: Dover Publications, 1950). Robert Kanigel, The Man Who Knew Infinity: A Life of the Genius Ramanujan
(New York: Charles Scribner's Sons, 1991). Jerry P. King, The Art of Mathematics (New York: Plenum Publishing
Corporation, 1992). Morris Kline, Mathematics: A Cultural Approach (Reading, Massachusetts:
Addison-Wesley Publishing Company, 1962). Morris Kline, Mathematical Thought from Ancient to Modern Times, Vol. 1
(New York: Oxford University Press, 1972). Konard Knopp, Infinite Sequences and Series (New York: Dover Publica
tions, 1956). Konard Knopp, Theory and Application of Infinite Series (New York: Dover
Publications, 1990). Stephan Korner, The Philosophy of Mathematics (New York: Dover Publica
tions, 1968). John Mcleish, Number (New York: Fawcett Columbine, 1991). Karl Menninger, Number Words and Number Symbols: A Cultural History of
Numbers (New York: Dover Publications, 1969). Michael Moffatt, The Ages of Mathematics, Vol. 1, The Origins (New York:
Doubleday & Company, 1977). Jane Muir, Of Men and Numbers (New York: Dodd, Mead & Company,
1961). Ernest Nagel and James R. Newman, Godel's Proof (New York: New York
University Press, 1958). James Newman, ed., The World of Mathematics (New York: Simon and
Schuster, 1956). Carroll Newsom, Mathematical Discourses (Englewood Cliffs, New Jersey:
Prentice-Hall,1964). Ivan Niven, Numbers: Rational and Irrational (Washington, DC: The Mathe
matical Association of America, 1961). C. Stanley Ogilvy and John T. Anderson, Excursions in Number Theory (New
York: Dover Publications, 1966). Oystein are, Number Theory and Its History (New York: Dover Publications,
1976). Theoni Pappas, The Joy of Mathematics (San Carlos, California: World Wide
Publishing/Tetra, 1989). John Allen Paulos, Innumeracy: Mathematical Illiteracy and Its Consequences
(New York: Hill and Wang, 1988). Rozsa Peter, Playing with Infinity (New York: Dover Publications, 1961).
308 MATHEMATICAL MYSTERIES
Ivars Peterson, Islands of Truth: A Mathematical Cruise (New York: w.H. Freeman and Company, 1990).
Plato, The Dialogues of Plato, trans. B. Jowett (New York: Random House, 1937).
Earl D. Rainville, Infinite Series (New York: The MacMillan Company, 1967). HL. Resnikoff and R.O. Wells, Jr., Mathematics in Civilization (New York:
Dover Publications, 1984). Paulo Ribenboim, The Book of Prime Number Records (New York: Springer
Verlag, 1989). Gay Robins and Charles Shute, The Rhind Mathematical Papyrus (New York:
Dover Publications, 1987). Rudy Rucker, Infinity and the Mind (New York: Bantam Books, 1982). Bertrand Russell, The Problems of Philosophy (London: Oxford University
Press, 1959). W. W. Sawyer, Mathematician's Delight (London: Penguin Books, 1943). Annemarie Schimmel, The Mystery of Numbers (Oxford: Oxford University
Press, 1993). Denise Schmandt-Besserat, Before Writing, Vol. I: From Counting to Cunei
form (Austin, Texas: University of Texas Press, 1992). David Eugene Smith, History of Mathematics, Vol. 1 (New York: Dover
Publications, 1951). Sherman K. Stein, Mathematics: The Man-made Universe (San Francisco: W.
H. Freeman and Company, 1963). Ian Stewart, The Problems of Mathematics (Oxford: Oxford University Press,
1987). Lloyd Strayhorn, Numbers and You (New York: Ballantine Books, 1987). Dirk J. Struik, A Concise History of Mathematics (New York: Dover Publica
tions, 1967). Frank J. Swetz, Capitalism and Arithmetic (La Salle, lllinois: Open Court
Publishing, 1987). Thomas Taylor, The Theoretic Arithmetic of the Pythagoreans (New York:
Samuel Weiser, 1972). David Wells, Curious and Interesting Numbers (London: Penguin Books, 1986). W.H. Werkmeister, A Philosophy of Science (Lincoln, Nebraska: University
of Nebraska Press, 1940). Alfred North Whitehead, An Introduction to Mathematics (New York: Ox
ford University Press, 1958). Guy Woodruff and David Premack, "Primate Mathematical Concepts in
the Chimpanzee: Proportionality and Numerosity," Nature 293 (October 15, 1981).
Claudia Zaslavsky, Africa Counts (New York: Lawrence Hill Books, 1973). Leo Zippin, Uses of Infinity (Washington, DC: The Mathematical Associa