# Pythagoras' Revenge: A Mathematical Mystery

By Arturo Sangalli | Go to book overview

Appendix 2
Infinitely Many Primes

Galway had come across Euclid’s proof that there are infinitely many prime numbers. Here’s a short primer on primes followed by the proof.

Some positive integers can be decomposed into a product of two smaller ones; 28 (= 4 × 7) and 315 (= 9 × 35) are examples of such composite numbers. Those that can’t be so decomposed are called prime. Put another way: a positive integer p > 1 is prime if its only (positive) divisors are itself and 1, so that it can only be decomposed in the trivial way p = 1 × p. Prime numbers are the “blocks” from which all numbers can be built, in the sense that every positive integer greater than 1 is a product of primes (or it is itself a prime)—and it is therefore divisible by some prime. For example, 4,095 = 3 × 3 × 5 × 7 × 13, so 4,095 is divisible by the primes 3, 5, 7, and 13.

The first ten prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, and 29. Does the list of primes eventually stop or does it go on forever? In other words, is the set of all prime numbers finite or infinite? Euclid answered this question in the Elements. He stated the theorem that there are infinitely many primes without using the term “infinity”: “Prime numbers are more than any given multitude of prime numbers” (Book IX, Proposition 20). His proof is short and beautifully simple. Here it is, essentially unchanged, in modern notation:

Consider the list of primes up to a certain prime P. If we multiply together all the numbers on the list and add 1 to the result, we obtain a number N = (2 × 3 × 5 ×… × P) + 1 greater than all those on the

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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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