# Pythagoras' Revenge: A Mathematical Mystery

By Arturo Sangalli | Go to book overview

Appendix 5
Perfect and Figured Numbers

Perfect Numbers

One classification of numbers attributed to the Pythagoreans involves comparing a number with the sum of its proper divisors—that is, excluding the number itself. If this sum is smaller than the number, the latter is called defective; if it is greater, the number is excessive. For example, 15 is defective, since 1 + 3 + 5 < 15, and 24 is excessive (24 < 1 + 2 + 3 + 4 + 6 + 8 + 12). A number is perfect if it’s neither defective nor excessive, in other words, if it is equal to the sum of its proper divisors. Examples of perfect numbers are 6 (= 1 + 2 + 3) and 28 (= 1 + 2 + 4 + 7 + 14).

In the final proposition of Book IX of his Elements, Euclid provided a recipe for constructing perfect numbers: if 2n − 1 is prime, then the product 2n−1 (2n − 1) is a perfect number. The two examples of perfect numbers above arise from Euclid’s formula by taking n = 2 and n = 3. The next two perfect numbers are 496 (for n = 5) and 8128 (n = 7).

If you were left with the impression that Euclid’s formula will readily lead to the discovery of plenty of perfect numbers, consider this: all the perfect numbers the Greeks ever knew were the four mentioned above, and it was not until the fifteenth century that another one was added to the list (212 (213 −1)). Today, only about forty perfect numbers are known, and their fascinating theory still contains many simple but unanswered questions, such as: Is there an odd perfect number? Are there infinitely many perfect numbers?

-178-

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Pythagoras' Revenge: A Mathematical Mystery

• Title Page iii
• Contents v
• Preface ix
• List of Main Characters (Chapter in Which They Are Introduced) xi
• Prologue xiii
• Part I- A Time Capsule? 1
• Chapter 1- The Fifteen Puzzle 3
• Chapter 2- The Impossible Manuscript 10
• Chapter 3- Game over 19
• Chapter 4- A Trip to London 25
• Chapter 5- A Letter from the Past 32
• Chapter 6- Found and Lost 38
• Chapter 7- A Death in the Family 46
• Part II- An Extraordinarily Gifted Man 51
• Chapter 8- The Mission 53
• Chapter 9- Norton Thorp 63
• Chapter 10- Random Numbers 69
• Chapter 11- Randomness Everywhere 76
• Chapter 12- Vanished 82
• Part III- A Sect of Neo­ Pythagoreans 83
• Chapter 13- The Mandate 85
• Chapter 14- The Beacon 87
• Chapter 15- The Team 98
• Chapter 16- The Hunt 106
• Chapter 17- The Symbol of the Serpent 115
• Chapter 18- A Professional Job 122
• Chapter 19- with a Little Help from Your Sister 126
• Part IV- Pythagoras' Mission 137
• Chapter 21- Kidnapped 152
• Chapter 22- The Last Piece of the Puzzle 158
• Epilogue 169
• Appendix 1- Jule's Solution 171
• Appendix 2- Infinitely Many Primes 173
• Appendix 3- Random Sequences 175
• Appendix 4- A Simple Visual Proof of the Pythagorean Theorem 177
• Appendix 5- Perfect and Figured Numbers 178
• Notes, Credits, and Bibliographical Sources 181
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