2013 UGA Math Camp Is there a pattern in the prime numbers? Is there a pattern in the prime numbers? Paul Pollack Intro A mystery? Paul Pollack So what is a pattern anyway? Randomness: A new hope La fin 1 / 74 Paul Pollack Is there a pattern in the prime numbers? June 18, 2013 Examples 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, . Theorem (Fundamental theorem of arithmetic) Every natural number > 1 can be written as a product of primes in exactly one way. For example, 2980398103281112391123 = 491 · 967 · 2767 · 2268595952977: The stars of our show Is there a Definition pattern in the prime A prime number is an integer > 1 whose only positive divisors numbers? Paul Pollack are 1 and itself. Intro A mystery? So what is a pattern anyway? Randomness: A new hope La fin 2 / 74 Paul Pollack Is there a pattern in the prime numbers? Theorem (Fundamental theorem of arithmetic) Every natural number > 1 can be written as a product of primes in exactly one way. For example, 2980398103281112391123 = 491 · 967 · 2767 · 2268595952977: The stars of our show Is there a Definition pattern in the prime A prime number is an integer > 1 whose only positive divisors numbers? Paul Pollack are 1 and itself. Intro Examples A mystery? 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, . So what is a pattern anyway? Randomness: A new hope La fin 3 / 74 Paul Pollack Is there a pattern in the prime numbers? The stars of our show Is there a Definition pattern in the prime A prime number is an integer > 1 whose only positive divisors numbers? Paul Pollack are 1 and itself. Intro Examples A mystery? 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, . So what is a pattern anyway? Randomness: Theorem (Fundamental theorem of arithmetic) A new hope Every natural number > 1 can be written as a product of La fin primes in exactly one way. For example, 2980398103281112391123 = 491 · 967 · 2767 · 2268595952977: 4 / 74 Paul Pollack Is there a pattern in the prime numbers? Couldn't we just list all of the prime numbers? Is there a pattern in the prime numbers? Paul Pollack Intro A mystery? So what is a pattern anyway? Randomness: A new hope Theorem (Euclid) La fin There are infinitely many primes. What Euclid's argument actually shows is that no matter what finite list of primes you start with, there's always at least one prime that you're missing. 5 / 74 Paul Pollack Is there a pattern in the prime numbers? Why? Well, p goes evenly into P , but none of the pi go evenly into P . In fact, we constructed P so that it was one more than a multiple of each pi. So if you divide P by pi, you'll get a remainder of 1 (not zero). 1ly many primes, ctd. Is there a pattern in the prime Proof. numbers? Paul Pollack Suppose you start with a finite list of primes, say p1; : : : ; pk. Let Intro P = p1 ··· p + 1: A mystery? k So what is a Certainly P > 1. So P can be factored into primes. Choose a pattern anyway? prime that shows up in the factorization of P , call it p. Randomness: A new hope Claim: p isn't one of p1; : : : ; pk. La fin 6 / 74 Paul Pollack Is there a pattern in the prime numbers? 1ly many primes, ctd. Is there a pattern in the prime Proof. numbers? Paul Pollack Suppose you start with a finite list of primes, say p1; : : : ; pk. Let Intro P = p1 ··· p + 1: A mystery? k So what is a Certainly P > 1. So P can be factored into primes. Choose a pattern anyway? prime that shows up in the factorization of P , call it p. Randomness: A new hope Claim: p isn't one of p1; : : : ; pk. La fin Why? Well, p goes evenly into P , but none of the pi go evenly into P . In fact, we constructed P so that it was one more than a multiple of each pi. So if you divide P by pi, you'll get a remainder of 1 (not zero). 7 / 74 Paul Pollack Is there a pattern in the prime numbers? 1ly many primes, ctd. Is there a pattern in the For example, suppose you want to find a prime not on the list prime numbers? 2; 3; 5; 7, and 11, and 13. We take Paul Pollack P = 2 · 3 · 5 · 7 · 11 + 1 Intro A mystery? = 30031: So what is a pattern anyway? Then any prime divisor of 30031 is a new prime not on our list. Randomness: It turns out 30031 = 59 · 509. So we can replace our old list A new hope with the list 2; 3; 5; 7, 11, 13, 59, 509. La fin We can repeat the argument as many times as desired to get a list of primes as long as desired. Moral: Since the sequence of primes continues forever, it makes sense to ask how it continues. In other words, how are 8 / 74 the primes distributed? Paul Pollack Is there a pattern in the prime numbers? In fact, one often hears that the prime numbers are randomly distributed, or seem to have no pattern. Can we make sense of these claims? Everyone loves a good mystery Is there a pattern in the prime numbers? Questions about the distribution of prime numbers are hard. Paul Pollack It's safe to say that prime numbers are some of the most Intro mysterious figures in all of mathematics. A mystery? So what is a pattern anyway? Randomness: A new hope La fin 9 / 74 Paul Pollack Is there a pattern in the prime numbers? Everyone loves a good mystery Is there a pattern in the prime numbers? Questions about the distribution of prime numbers are hard. Paul Pollack It's safe to say that prime numbers are some of the most Intro mysterious figures in all of mathematics. A mystery? So what is a pattern anyway? In fact, one often hears that the prime Randomness: A new hope numbers are randomly distributed, or seem to La fin have no pattern. Can we make sense of these claims? 10 / 74 Paul Pollack Is there a pattern in the prime numbers? 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 Only two of the 6 columns contain primes. 6n + 2 = 2(3n + 1), 6n + 3 = 3(2n + 1), 6n + 4 = 2(3n + 2), and 6n + 6 = 6(n + 1). Impressed yet? Psychic math Is there a pattern in the prime numbers? 1 2 Paul Pollack 3 4 5 6 Intro A mystery? 7 8 So what is a 9 10 pattern anyway? 11 12 Randomness: 13 14 A new hope 15 16 La fin 17 18 11 / 74 Paul Pollack Is there a pattern in the prime numbers? Only two of the 6 columns contain primes. 6n + 2 = 2(3n + 1), 6n + 3 = 3(2n + 1), 6n + 4 = 2(3n + 2), and 6n + 6 = 6(n + 1). Impressed yet? Psychic math Is there a pattern in the prime numbers? 1 2 1 2 3 4 5 6 Paul Pollack 3 4 7 8 9 10 11 12 5 6 13 14 15 16 17 18 Intro A mystery? 7 8 19 20 21 22 23 24 So what is a 9 10 25 26 27 28 29 30 pattern anyway? 11 12 31 32 33 34 35 36 Randomness: 13 14 37 38 39 40 41 42 A new hope 15 16 43 44 45 46 47 48 La fin 17 18 49 50 51 52 53 54 12 / 74 Paul Pollack Is there a pattern in the prime numbers? Psychic math Is there a pattern in the prime numbers? 1 2 1 2 3 4 5 6 Paul Pollack 3 4 7 8 9 10 11 12 5 6 13 14 15 16 17 18 Intro A mystery? 7 8 19 20 21 22 23 24 So what is a 9 10 25 26 27 28 29 30 pattern anyway? 11 12 31 32 33 34 35 36 Randomness: 13 14 37 38 39 40 41 42 A new hope 15 16 43 44 45 46 47 48 La fin 17 18 49 50 51 52 53 54 Only two of the 6 columns contain primes. 6n + 2 = 2(3n + 1), 6n + 3 = 3(2n + 1), 6n + 4 = 2(3n + 2), and 6n + 6 = 6(n + 1). Impressed yet? 13 / 74 Paul Pollack Is there a pattern in the prime numbers? But that's not true either! Example (Sierpi´nski) Consider the constant A = 0:02030005000000070::: defined by 1 X −2k pk · 10 ; k=1 where p1 = 2, p2 = 3, p3 = 5, . Then 2k 2k−1 2k−1 pk = b10 Ac − 10 b10 Ac: What is a pattern? Is there a pattern in the Maybe \No pattern" means \no formula"? prime numbers? Paul Pollack Intro A mystery? So what is a pattern anyway? Randomness: A new hope La fin 14 / 74 Paul Pollack Is there a pattern in the prime numbers? What is a pattern? Is there a pattern in the Maybe \No pattern" means \no formula"? prime numbers? But that's not true either! Paul Pollack Example (Sierpi´nski) Intro A mystery? Consider the constant So what is a pattern A = 0:02030005000000070::: anyway? Randomness: A new hope defined by 1 La fin X −2k pk · 10 ; k=1 where p1 = 2, p2 = 3, p3 = 5, .