# Pythagoras' Revenge: A Mathematical Mystery

By Arturo Sangalli | Go to book overview

Appendix 3
Random Sequences

In 1919, the Austrian-born mathematician Richard von Mises proposed the following definition: an infinite sequence s, say, of os and 1s is random if

(a) s satisfies the law of large numbers, that is, ‘there are as many os as there are 1s,’ or, more precisely, the limiting value of x/n, where x is the number of os among the first n terms of the sequence, is 0.5, and

(b) Every subsequence that can be extracted from s by reasonable means also satisfies the law of large numbers.

Applying von Mises definition to the sequence 0 1 0 1 0 1 0 1… of alternating 0s and 1s would confirm that it is not random, for the subsequence of even bits (the 2nd, 4th, etc.), that is, 1 1 1 1 1 1…, clearly does not satisfy condition (a). Likewise, many other sequences which appear intuitively to be nonrandom fail to satisfy von Mises’ conditions, and hence they are not random also in the technical sense.

Unfortunately, the definition proposed by the Austrian mathematician suffered from a fundamental defect: it did not specify which means for extracting a subsequence are “reasonable” means. To remedy this situation Alonzo Church, an American mathematician, suggested in 1940 that condition (b) of von Mises’ definition should only apply to computable subsequences, that is, to those subsequences whose terms could be defined by a computer program. Although Church’s idea had the merit of making the definition precise, examples were subsequently found of sequences that are intuitively

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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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