Is There a Pattern in the Prime Numbers? Is There a Pattern in the Prime Numbers? Paul Pollack

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).

∞ly many primes, ctd.

6 / 74 Paul Pollack Is there a pattern in the prime numbers? ∞ly 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? ∞ly 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

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 ∞ 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 ∞ La ﬁn 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. 15 / 74 Paul Pollack Is there a pattern in the prime numbers? 43 + 4 = 47, which is prime. 47 + 6 = 53, which is prime. 53 + 8 = 61, which is prime. 61 + 10 = 71, which is prime. 71 + 12 = 83, which is prime. 83 + 14 = 97, which is prime. 97 + 16 = 113, which is prime. 113 + 18 = 131, which is prime. 131 + 20 = 151, which is prime. 151 + 22 = 173, which is prime. 173 + 24 = 197, which is prime. 197 + 26 = 223, which is prime.

OK, how about a useful formula?

Is there a pattern in the prime Let’s start with some observations of Leonhard Euler. numbers? Paul Pollack 41 + 2 = 43, which is prime.

Intro

A mystery?

So what is a pattern anyway?

Randomness: A new hope

La ﬁn

16 / 74 Paul Pollack Is there a pattern in the prime numbers? 47 + 6 = 53, which is prime. 53 + 8 = 61, which is prime. 61 + 10 = 71, which is prime. 71 + 12 = 83, which is prime. 83 + 14 = 97, which is prime. 97 + 16 = 113, which is prime. 113 + 18 = 131, which is prime. 131 + 20 = 151, which is prime. 151 + 22 = 173, which is prime. 173 + 24 = 197, which is prime. 197 + 26 = 223, which is prime.

OK, how about a useful formula?

Is there a pattern in the prime Let’s start with some observations of Leonhard Euler. numbers? Paul Pollack 41 + 2 = 43, which is prime. 43 + 4 = 47, which is prime. Intro

A mystery?

So what is a pattern anyway?

Randomness: A new hope

La ﬁn

17 / 74 Paul Pollack Is there a pattern in the prime numbers? 53 + 8 = 61, which is prime. 61 + 10 = 71, which is prime. 71 + 12 = 83, which is prime. 83 + 14 = 97, which is prime. 97 + 16 = 113, which is prime. 113 + 18 = 131, which is prime. 131 + 20 = 151, which is prime. 151 + 22 = 173, which is prime. 173 + 24 = 197, which is prime. 197 + 26 = 223, which is prime.

OK, how about a useful formula?

Is there a pattern in the prime Let’s start with some observations of Leonhard Euler. numbers? Paul Pollack 41 + 2 = 43, which is prime. 43 + 4 = 47, which is prime. Intro

A mystery? 47 + 6 = 53, which is prime.

So what is a pattern anyway?

Randomness: A new hope

La ﬁn

18 / 74 Paul Pollack Is there a pattern in the prime numbers? 61 + 10 = 71, which is prime. 71 + 12 = 83, which is prime. 83 + 14 = 97, which is prime. 97 + 16 = 113, which is prime. 113 + 18 = 131, which is prime. 131 + 20 = 151, which is prime. 151 + 22 = 173, which is prime. 173 + 24 = 197, which is prime. 197 + 26 = 223, which is prime.

OK, how about a useful formula?

Is there a pattern in the prime Let’s start with some observations of Leonhard Euler. numbers? Paul Pollack 41 + 2 = 43, which is prime. 43 + 4 = 47, which is prime. Intro

A mystery? 47 + 6 = 53, which is prime.

So what is a 53 + 8 = 61, which is prime. pattern anyway?

Randomness: A new hope

La ﬁn

19 / 74 Paul Pollack Is there a pattern in the prime numbers? 71 + 12 = 83, which is prime. 83 + 14 = 97, which is prime. 97 + 16 = 113, which is prime. 113 + 18 = 131, which is prime. 131 + 20 = 151, which is prime. 151 + 22 = 173, which is prime. 173 + 24 = 197, which is prime. 197 + 26 = 223, which is prime.

OK, how about a useful formula?

Is there a pattern in the prime Let’s start with some observations of Leonhard Euler. numbers? Paul Pollack 41 + 2 = 43, which is prime. 43 + 4 = 47, which is prime. Intro

A mystery? 47 + 6 = 53, which is prime.

So what is a 53 + 8 = 61, which is prime. pattern anyway? 61 + 10 = 71, which is prime.

Randomness: A new hope

La ﬁn

20 / 74 Paul Pollack Is there a pattern in the prime numbers? 83 + 14 = 97, which is prime. 97 + 16 = 113, which is prime. 113 + 18 = 131, which is prime. 131 + 20 = 151, which is prime. 151 + 22 = 173, which is prime. 173 + 24 = 197, which is prime. 197 + 26 = 223, which is prime.

OK, how about a useful formula?

Is there a pattern in the prime Let’s start with some observations of Leonhard Euler. numbers? Paul Pollack 41 + 2 = 43, which is prime. 43 + 4 = 47, which is prime. Intro

A mystery? 47 + 6 = 53, which is prime.

So what is a 53 + 8 = 61, which is prime. pattern anyway? 61 + 10 = 71, which is prime.

Randomness: 71 + 12 = 83, which is prime. A new hope

La ﬁn

21 / 74 Paul Pollack Is there a pattern in the prime numbers? 97 + 16 = 113, which is prime. 113 + 18 = 131, which is prime. 131 + 20 = 151, which is prime. 151 + 22 = 173, which is prime. 173 + 24 = 197, which is prime. 197 + 26 = 223, which is prime.

OK, how about a useful formula?

Is there a pattern in the prime Let’s start with some observations of Leonhard Euler. numbers? Paul Pollack 41 + 2 = 43, which is prime. 43 + 4 = 47, which is prime. Intro

A mystery? 47 + 6 = 53, which is prime.

So what is a 53 + 8 = 61, which is prime. pattern anyway? 61 + 10 = 71, which is prime.

Randomness: 71 + 12 = 83, which is prime. A new hope 83 + 14 = 97, which is prime. La ﬁn

22 / 74 Paul Pollack Is there a pattern in the prime numbers? 113 + 18 = 131, which is prime. 131 + 20 = 151, which is prime. 151 + 22 = 173, which is prime. 173 + 24 = 197, which is prime. 197 + 26 = 223, which is prime.

OK, how about a useful formula?

Is there a pattern in the prime Let’s start with some observations of Leonhard Euler. numbers? Paul Pollack 41 + 2 = 43, which is prime. 43 + 4 = 47, which is prime. Intro

A mystery? 47 + 6 = 53, which is prime.

So what is a 53 + 8 = 61, which is prime. pattern anyway? 61 + 10 = 71, which is prime.

Randomness: 71 + 12 = 83, which is prime. A new hope 83 + 14 = 97, which is prime. La ﬁn 97 + 16 = 113, which is prime.

23 / 74 Paul Pollack Is there a pattern in the prime numbers? 131 + 20 = 151, which is prime. 151 + 22 = 173, which is prime. 173 + 24 = 197, which is prime. 197 + 26 = 223, which is prime.

OK, how about a useful formula?

Is there a pattern in the prime Let’s start with some observations of Leonhard Euler. numbers? Paul Pollack 41 + 2 = 43, which is prime. 43 + 4 = 47, which is prime. Intro

A mystery? 47 + 6 = 53, which is prime.

So what is a 53 + 8 = 61, which is prime. pattern anyway? 61 + 10 = 71, which is prime.

Randomness: 71 + 12 = 83, which is prime. A new hope 83 + 14 = 97, which is prime. La ﬁn 97 + 16 = 113, which is prime. 113 + 18 = 131, which is prime.

24 / 74 Paul Pollack Is there a pattern in the prime numbers? 151 + 22 = 173, which is prime. 173 + 24 = 197, which is prime. 197 + 26 = 223, which is prime.

OK, how about a useful formula?

Is there a pattern in the prime Let’s start with some observations of Leonhard Euler. numbers? Paul Pollack 41 + 2 = 43, which is prime. 43 + 4 = 47, which is prime. Intro

A mystery? 47 + 6 = 53, which is prime.

So what is a 53 + 8 = 61, which is prime. pattern anyway? 61 + 10 = 71, which is prime.

Randomness: 71 + 12 = 83, which is prime. A new hope 83 + 14 = 97, which is prime. La ﬁn 97 + 16 = 113, which is prime. 113 + 18 = 131, which is prime. 131 + 20 = 151, which is prime.

25 / 74 Paul Pollack Is there a pattern in the prime numbers? 173 + 24 = 197, which is prime. 197 + 26 = 223, which is prime.

OK, how about a useful formula?

Is there a pattern in the prime Let’s start with some observations of Leonhard Euler. numbers? Paul Pollack 41 + 2 = 43, which is prime. 43 + 4 = 47, which is prime. Intro

A mystery? 47 + 6 = 53, which is prime.

So what is a 53 + 8 = 61, which is prime. pattern anyway? 61 + 10 = 71, which is prime.

Randomness: 71 + 12 = 83, which is prime. A new hope 83 + 14 = 97, which is prime. La ﬁn 97 + 16 = 113, which is prime. 113 + 18 = 131, which is prime. 131 + 20 = 151, which is prime. 151 + 22 = 173, which is prime.

26 / 74 Paul Pollack Is there a pattern in the prime numbers? 197 + 26 = 223, which is prime.

OK, how about a useful formula?

Is there a pattern in the prime Let’s start with some observations of Leonhard Euler. numbers? Paul Pollack 41 + 2 = 43, which is prime. 43 + 4 = 47, which is prime. Intro

A mystery? 47 + 6 = 53, which is prime.

So what is a 53 + 8 = 61, which is prime. pattern anyway? 61 + 10 = 71, which is prime.

27 / 74 Paul Pollack Is there a pattern in the prime numbers? OK, how about a useful formula?

Is there a pattern in the prime Let’s start with some observations of Leonhard Euler. numbers? Paul Pollack 41 + 2 = 43, which is prime. 43 + 4 = 47, which is prime. Intro

A mystery? 47 + 6 = 53, which is prime.

So what is a 53 + 8 = 61, which is prime. pattern anyway? 61 + 10 = 71, which is prime.

28 / 74 197 + 26 = 223, which is prime. Paul Pollack Is there a pattern in the prime numbers? Theorem For almost all (asymptotically 100%) natural numbers n, the number n2 + n + 41 is composite.

If we plug in n = 40, we get

41 + 402 + 40 = 41 + 40(40 + 1) = 41 + 40 · 41 = 412,

which is composite.

Things fall apart

Is there a pattern in the The nth number output by this process is prime numbers? Paul Pollack 41 + 2 + 4 + 6 + ··· + 2n Intro = 41 + 2(1 + 2 + 3 + ··· + n) = 41 + n2 + n. A mystery?

So what is a pattern anyway?

Randomness: A new hope

La ﬁn

29 / 74 Paul Pollack Is there a pattern in the prime numbers? Things fall apart

Is there a pattern in the The nth number output by this process is prime numbers? Paul Pollack 41 + 2 + 4 + 6 + ··· + 2n Intro = 41 + 2(1 + 2 + 3 + ··· + n) = 41 + n2 + n. A mystery?

So what is a pattern anyway? If we plug in n = 40, we get

Randomness: 2 2 A new hope 41 + 40 + 40 = 41 + 40(40 + 1) = 41 + 40 · 41 = 41 , La ﬁn which is composite. Theorem For almost all (asymptotically 100%) natural numbers n, the number n2 + n + 41 is composite. 30 / 74 Paul Pollack Is there a pattern in the prime numbers? 15, 825, 725, 1925, 2275, 425, 390, 330, 290, 770, 910, 170, 156, 132, 116, 308, 364, 68, 4,....

Primes through fractions

Is there a pattern in the prime The following is due to the contemporary mathematician John numbers? Conway. Consider the following list of fourteen fractions: Paul Pollack

17 78 19 23 29 77 95 77 1 11 13 15 15 55 Intro 91 85 51 38 33 29 23 19 17 13 11 14 2 1 A mystery? We start by setting n = 2. To continue ... So what is a pattern anyway? * Find the ﬁrst fraction in the list which you can multiply n Randomness: by to get an integer; replace n with that new integer. A new hope k La ﬁn Whenever you reach a power of 2, say 2 , output the exponent k. Now repeat.

Going through this process, we run through the integers

31 / 74 Paul Pollack Is there a pattern in the prime numbers? 825, 725, 1925, 2275, 425, 390, 330, 290, 770, 910, 170, 156, 132, 116, 308, 364, 68, 4,....

Primes through fractions

Is there a pattern in the prime The following is due to the contemporary mathematician John numbers? Conway. Consider the following list of fourteen fractions: Paul Pollack

Going through this process, we run through the integers 15,

32 / 74 Paul Pollack Is there a pattern in the prime numbers? 725, 1925, 2275, 425, 390, 330, 290, 770, 910, 170, 156, 132, 116, 308, 364, 68, 4,....

Primes through fractions

Is there a pattern in the prime The following is due to the contemporary mathematician John numbers? Conway. Consider the following list of fourteen fractions: Paul Pollack

Going through this process, we run through the integers 15, 825,

33 / 74 Paul Pollack Is there a pattern in the prime numbers? 1925, 2275, 425, 390, 330, 290, 770, 910, 170, 156, 132, 116, 308, 364, 68, 4,....

Primes through fractions

Is there a pattern in the prime The following is due to the contemporary mathematician John numbers? Conway. Consider the following list of fourteen fractions: Paul Pollack

Going through this process, we run through the integers 15, 825, 725,

34 / 74 Paul Pollack Is there a pattern in the prime numbers? 2275, 425, 390, 330, 290, 770, 910, 170, 156, 132, 116, 308, 364, 68, 4,....

Primes through fractions

Is there a pattern in the prime The following is due to the contemporary mathematician John numbers? Conway. Consider the following list of fourteen fractions: Paul Pollack

Going through this process, we run through the integers 15, 825, 725, 1925,

35 / 74 Paul Pollack Is there a pattern in the prime numbers? Primes through fractions

Is there a pattern in the prime The following is due to the contemporary mathematician John numbers? Conway. Consider the following list of fourteen fractions: Paul Pollack

Going through this process, we run through the integers 15, 825, 725, 1925, 2275, 425, 390, 330, 290, 770, 910, 170, 156, 132, 116, 308, 364, 68, 4,.... 36 / 74 Paul Pollack Is there a pattern in the prime numbers? A recent photograph of John Conway

Is there a pattern in the prime numbers?

Paul Pollack

Intro

A mystery?

So what is a pattern anyway?

Randomness: A new hope

La ﬁn

37 / 74 Paul Pollack Is there a pattern in the prime numbers? 3 69 5 281 7 710 11 2375 13 3893 17 8102

Theorem (Conway) The set of outputs is exactly the set of primes, in increasing order.

Primes through fractions, ctd.

Is there a pattern in the k prime Remember that every time we see 2 , we output the value of k. numbers? Paul Pollack Output Number of steps required

Intro 2 19

A mystery?

So what is a pattern anyway?

Randomness: A new hope

La ﬁn

38 / 74 Paul Pollack Is there a pattern in the prime numbers? 3 69 5 281 7 710 11 2375 13 3893 17 8102

Theorem (Conway) The set of outputs is exactly the set of primes, in increasing order.

Primes through fractions, ctd.

Is there a pattern in the k prime Remember that every time we see 2 , we output the value of k. numbers? Paul Pollack Output Number of steps required

Intro 2 19

A mystery?

So what is a pattern anyway?

Randomness: A new hope

La ﬁn

39 / 74 Paul Pollack Is there a pattern in the prime numbers? 5 281 7 710 11 2375 13 3893 17 8102

Theorem (Conway) The set of outputs is exactly the set of primes, in increasing order.

Primes through fractions, ctd.

Is there a pattern in the k prime Remember that every time we see 2 , we output the value of k. numbers? Paul Pollack Output Number of steps required

Intro 2 19

A mystery? 3 69

So what is a pattern anyway?

Randomness: A new hope

La ﬁn

40 / 74 Paul Pollack Is there a pattern in the prime numbers? 7 710 11 2375 13 3893 17 8102

Theorem (Conway) The set of outputs is exactly the set of primes, in increasing order.

Primes through fractions, ctd.

Is there a pattern in the k prime Remember that every time we see 2 , we output the value of k. numbers? Paul Pollack Output Number of steps required

Intro 2 19

A mystery? 3 69

So what is a 5 281 pattern anyway?

Randomness: A new hope

La ﬁn

41 / 74 Paul Pollack Is there a pattern in the prime numbers? 11 2375 13 3893 17 8102

Theorem (Conway) The set of outputs is exactly the set of primes, in increasing order.

Primes through fractions, ctd.

Is there a pattern in the k prime Remember that every time we see 2 , we output the value of k. numbers? Paul Pollack Output Number of steps required

Intro 2 19

A mystery? 3 69

So what is a 5 281 pattern anyway? 7 710

Randomness: A new hope

La ﬁn

42 / 74 Paul Pollack Is there a pattern in the prime numbers? 13 3893 17 8102

Theorem (Conway) The set of outputs is exactly the set of primes, in increasing order.

Primes through fractions, ctd.

Is there a pattern in the k prime Remember that every time we see 2 , we output the value of k. numbers? Paul Pollack Output Number of steps required

Intro 2 19

A mystery? 3 69

So what is a 5 281 pattern anyway? 7 710

Randomness: 11 2375 A new hope

La ﬁn

43 / 74 Paul Pollack Is there a pattern in the prime numbers? 17 8102

Theorem (Conway) The set of outputs is exactly the set of primes, in increasing order.

Primes through fractions, ctd.

Is there a pattern in the k prime Remember that every time we see 2 , we output the value of k. numbers? Paul Pollack Output Number of steps required

Intro 2 19

A mystery? 3 69

So what is a 5 281 pattern anyway? 7 710

Randomness: 11 2375 A new hope 13 3893 La ﬁn

44 / 74 Paul Pollack Is there a pattern in the prime numbers? Theorem (Conway) The set of outputs is exactly the set of primes, in increasing order.

Primes through fractions, ctd.

Is there a pattern in the k prime Remember that every time we see 2 , we output the value of k. numbers? Paul Pollack Output Number of steps required

Intro 2 19

A mystery? 3 69

So what is a 5 281 pattern anyway? 7 710

Randomness: 11 2375 A new hope 13 3893 La ﬁn 17 8102

45 / 74 Paul Pollack Is there a pattern in the prime numbers? Primes through fractions, ctd.

Is there a pattern in the k prime Remember that every time we see 2 , we output the value of k. numbers? Paul Pollack Output Number of steps required

Intro 2 19

A mystery? 3 69

So what is a 5 281 pattern anyway? 7 710

Randomness: 11 2375 A new hope 13 3893 La ﬁn 17 8102

Theorem (Conway) The set of outputs is exactly the set of primes, in increasing order.

46 / 74 Paul Pollack Is there a pattern in the prime numbers? Another way of generating the primes in increasing order

Is there a pattern in the prime numbers? Paul Pollack It’s easy to write a computer program to generate the primes. Intro A (not so great) algorithm for generating the primes: A mystery?

So what is a pattern Run through the integers n = 1, 2, 3,... in order, anyway? outputting the number if only if it’s prime. Randomness: A new hope To test if n is prime, divide n by n − 1, n − 2,..., 1 La ﬁn successively until something goes in evenly. If we ﬁnd a divisor before we get down to 1, then n is composite. Otherwise, n is prime.

47 / 74 Paul Pollack Is there a pattern in the prime numbers? Another way of generating the primes in increasing order

Is there a pattern in the prime Actually this is exactly what we were doing before. Instead of numbers? programming in C or Pascal or Java, we were programming in a Paul Pollack language based on fractions, called FRACTRAN. Intro A mystery? Theorem (Conway) So what is a pattern For any sequence of positive integers that can be output by a anyway? computer program, there is a ﬁnite list of fractions which will Randomness: A new hope generate this sequence.

La ﬁn This is an amazingly general theorem. Unfortunately, precisely because it’s so general, it’s not reasonable to hope that it tell us anything interesting about the primes in particular. 48 / 74 Paul Pollack Is there a pattern in the prime numbers? But there’s another way in which primes behave randomly that I think is enlightening. Before doing that, I need to summarize some of what modern mathematicians have shown about the distribution of the prime numbers. Recall our notation π(x) for the number of primes not exceeding x.

Cutting our losses

Is there a pattern in the prime numbers? At this point, maybe you (like me) are of the opinion that a Paul Pollack useful formula for primes isn’t likely to crop up. Intro

A mystery? Saying that no such formula exists is one way to make sense of

So what is a the idea that “primes are patternless”. pattern anyway?

Randomness: A new hope

La ﬁn

49 / 74 Paul Pollack Is there a pattern in the prime numbers? Cutting our losses

Is there a pattern in the prime numbers? At this point, maybe you (like me) are of the opinion that a Paul Pollack useful formula for primes isn’t likely to crop up. Intro

A mystery? Saying that no such formula exists is one way to make sense of

So what is a the idea that “primes are patternless”. pattern anyway? But there’s another way in which primes behave randomly that Randomness: A new hope I think is enlightening. Before doing that, I need to summarize

La ﬁn some of what modern mathematicians have shown about the distribution of the prime numbers. Recall our notation π(x) for the number of primes not exceeding x.

50 / 74 Paul Pollack Is there a pattern in the prime numbers? While a young teenager, Gauss became interested in understanding the rate at which the primes thinned out. He reports that he often spent an idle quarter of an hour to count another chiliad here and there.

102 25 103 168 104 1,229 105 9,592 106 78,498 107 664,579 108 5,761,455 109 50,847534 1010 455,052,511 1011 4,118,054,813 1012 37,607,912,018 1013 346,065,536,839 1014 3,204,941,750,802

The counts of primes up to x, for x = 10k

x π(x) Is there a pattern in the prime 10 4 numbers?

Paul Pollack

Intro

A mystery?

So what is a pattern anyway?

Randomness: A new hope

La ﬁn

51 / 74 Paul Pollack Is there a pattern in the prime numbers? 103 168 104 1,229 105 9,592 While a young teenager, Gauss 106 78,498 became interested in 107 664,579 understanding the rate at 108 5,761,455 which the primes thinned out. 109 50,847534 He reports that he often 1010 455,052,511 spent an idle quarter 1011 4,118,054,813 of an hour to count 1012 37,607,912,018 another chiliad here 1013 346,065,536,839 and there. 1014 3,204,941,750,802

The counts of primes up to x, for x = 10k

x π(x) Is there a pattern in the prime 10 4 numbers? 102 25 Paul Pollack

Intro

A mystery?

So what is a pattern anyway?

Randomness: A new hope

La ﬁn

52 / 74 Paul Pollack Is there a pattern in the prime numbers? 104 1,229 105 9,592 While a young teenager, Gauss 106 78,498 became interested in 107 664,579 understanding the rate at 108 5,761,455 which the primes thinned out. 109 50,847534 He reports that he often 1010 455,052,511 spent an idle quarter 1011 4,118,054,813 of an hour to count 1012 37,607,912,018 another chiliad here 1013 346,065,536,839 and there. 1014 3,204,941,750,802

The counts of primes up to x, for x = 10k

x π(x) Is there a pattern in the prime 10 4 numbers? 102 25 Paul Pollack 103 168

Intro

A mystery?

So what is a pattern anyway?

Randomness: A new hope

La ﬁn

53 / 74 Paul Pollack Is there a pattern in the prime numbers? The counts of primes up to x, for x = 10k

x π(x) Is there a pattern in the prime 10 4 numbers? 102 25 Paul Pollack 103 168 4 Intro 10 1,229 5 A mystery? 10 9,592 While a young teenager, Gauss So what is a 106 78,498 became interested in pattern 7 anyway? 10 664,579 understanding the rate at Randomness: 108 5,761,455 which the primes thinned out. A new hope 9 He reports that he often La ﬁn 10 50,847534 1010 455,052,511 spent an idle quarter 1011 4,118,054,813 of an hour to count 1012 37,607,912,018 another chiliad here 1013 346,065,536,839 and there. 1014 3,204,941,750,802 54 / 74 Paul Pollack Is there a pattern in the prime numbers? Gauss’s guess about π(x)

Is there a pattern in the These investigations led to the following conjecture. prime numbers? Conjecture Paul Pollack The number of primes p ≤ x is approximately Intro x A mystery? , So what is a ln x pattern anyway? where ln x is the natural logarithm of x. Here “approximately” Randomness: A new hope means that the relative error you make if you approximate π(x)

La ﬁn by x/ ln x tends to 0 as x → ∞.

Note: If you haven’t seen the function ln before, you can pretend that

ln(x) = 1 + 1/2 + 1/3 + ··· + 1/bxc. 55 / 74 Paul Pollack Is there a pattern in the prime numbers? Testing Gauss’s guess

Is there a pattern in the prime π(x)−x/ ln x numbers? By the relative error, we mean the ratio π(x) . Paul Pollack x π(x) relative error Intro 103 168 13.7% A mystery? 104 1,229 11.6% So what is a 5 pattern 10 9,592 9.4% anyway? 106 78,498 7.8% Randomness: 7 A new hope 10 664,579 6.6% La ﬁn 108 5,761,455 5.8% 109 50,847534 5.1% 1010 455,052,511 4.6% 1011 4,118,054,813 4.1& 1012 37,607,912,018 3.8%

56 / 74 Paul Pollack Is there a pattern in the prime numbers? Testing Gauss’s guess, ctd.

Is there a pattern in the prime numbers?

Paul Pollack So Gauss’s guess is looking pretty good. Gauss also suggested Intro what he thought was an even better approximation to π(x), A mystery? the function x So what is a Z 1 pattern li(x) := dt. anyway? 2 ln t Randomness: A new hope It turns out that li(x) and x/ ln x are close enough for large x La ﬁn that if one of them is a good approximation to π(x) — in the sense of Gauss’s guess — then the other is too.

57 / 74 Paul Pollack Is there a pattern in the prime numbers? Comparing π(x), x/ ln x, and li(x) up to 1000

Is there a pattern in the prime numbers?

Paul Pollack

Intro 150

A mystery?

So what is a pattern 100 anyway?

Randomness: A new hope p X 50 La ﬁn X logH XL li X 200 400 600 800 1000 H L

H L

58 / 74 Paul Pollack Is there a pattern in the prime numbers? Comparing π(x), x/ ln x, and li(x) up to 100000

Is there a pattern in the prime numbers?

Paul Pollack

Intro 8000

A mystery?

So what is a 6000 pattern anyway? 4000 Randomness: p X A new hope La ﬁn 2000 X logH XL li X 20 000 40 000 60 000 80 000 100 000 H L

H L

59 / 74 Paul Pollack Is there a pattern in the prime numbers? OK, great, but what does this have to do with randomness? It turns out that Gauss’s guess about the size of π(x) is exactly what one would expect if primes behave randomly, in a sense we’ll now make precise.

Vindication at last

Is there a pattern in the prime About a hundred years after Gauss made his guess, two numbers? mathematicians — working independently but almost Paul Pollack simultaneously — conﬁrmed that Gauss was correct. Intro A mystery? Prime number theorem (Hadamard and de la So what is a Vallee–Poussin) pattern anyway? Gauss’s guess was correct. The relative error between π(x) and Randomness: A new hope x/ ln x (or between π(x) and li(x)) tends to zero as x → ∞.

La ﬁn

60 / 74 Paul Pollack Is there a pattern in the prime numbers? Vindication at last

La ﬁn OK, great, but what does this have to do with randomness? It turns out that Gauss’s guess about the size of π(x) is exactly what one would expect if primes behave randomly, in a sense we’ll now make precise. 61 / 74 Paul Pollack Is there a pattern in the prime numbers? The sieve of Eratosthenes

Is there a pattern in the prime numbers?

Paul Pollack

Intro The following method, discovered by the ancient A mystery? Greek Eratosthenes, is a useful method for generating all of the So what is a primes up to a certain height N. pattern anyway? Randomness: 1 Start with a list of all of the natural numbers from 2 to N. A new hope

La ﬁn 2 Look for the ﬁrst natural number that hasn’t been crossed out. This number is a prime; mark it as such. Now cross out all of its remaining multiples. Repeat this step.

The algorithm continues until all of the numbers are marked as prime or crossed out. 62 / 74 Paul Pollack Is there a pattern in the prime numbers? The sieve of Eratosthenes, ctd.

Is there a pattern in the prime If you want the entire sequence of primes, you can run this numbers?

Paul Pollack process with the interval from 2 to N replaced by the entire set of integers at least 2. More formally: Intro

A mystery? Let A1 = {2, 3, 4,... }. Let p1 be the first element of A1 So what is a pattern (so p1 = 2). Remove from A1 all of the elements are anyway? divisible by p1, and call the resulting set A2. Randomness: A new hope Let p2 be the first element of A2 (so p2 = 3). Remove La fin from A2 all of its remaining elements that are divisible by p2. Call this A3. Continue!

Then the sequence of primes is precisely p1, p2, p3,... . 63 / 74 Paul Pollack Is there a pattern in the prime numbers? Let A1 = {2, 3, 4,... }. Let p1 be the ﬁrst element of A1 (so p1 = 2). Remove p1 from A1, and remove each remaining element with probability 1/p1. Call the resulting set A2.

Let p2 be the ﬁrst element of A2. Remove p2 from A2, and then remove each remaining element with probability 1/p2. Call this A3. Continue!

There are then inﬁnitely many possibilities for the resulting sequence p1, p2, p3,... .

A random sieve

Is there a pattern in the The sieve of Eratosthenes is completely deterministic. Let’s prime numbers? introduce an element of chance into the picture. (This idea is Paul Pollack due to David Hawkins.)

Intro

A mystery?

So what is a pattern anyway?

Randomness: A new hope

La ﬁn

64 / 74 Paul Pollack Is there a pattern in the prime numbers? A random sieve

Intro Let A1 = {2, 3, 4,... }. Let p1 be the ﬁrst element of A1 A mystery? (so p = 2). Remove p from A , and remove each So what is a 1 1 1 pattern remaining element with probability 1/p . Call the resulting anyway? 1

Randomness: set A2. A new hope Let p2 be the ﬁrst element of A2. Remove p2 from A2, La ﬁn and then remove each remaining element with probability 1/p2. Call this A3. Continue!

There are then inﬁnitely many possibilities for the resulting

65 / 74 sequence p1, p2, p3,... . Paul Pollack Is there a pattern in the prime numbers? It’s 1 2 , since 3 has to not be killed by 2, and being killed by 2 1 happens with probability 2 .

Exercise: What’s the probability 4 ends up in our R-prime sequence? Now there are two cases, depending on whether 3 is there too or not.

A random sieve, ctd.

Is there a pattern in the prime There are then inﬁnitely many possibilities for the resulting numbers? sequence p1, p2, p3,... . We’ll call this an R-primes sequence. Paul Pollack Given a property of an inﬁnite sequence, we can ask: What’s Intro the probability that a random R-primes sequence has the A mystery? property. So what is a pattern anyway? Example Randomness: A new hope What’s the probability that 3 is in our R-prime sequence?

La ﬁn

66 / 74 Paul Pollack Is there a pattern in the prime numbers? A random sieve, ctd.

Exercise: What’s the probability 4 ends up in our R-prime sequence? Now there are two cases, depending on whether 3 is

67 / 74 there too or not. Paul Pollack Is there a pattern in the prime numbers? Theorem (Wunderlich) The probability that Gauss’s guess about π(x) is true for a random R-prime sequence is 100%.

If we didn’t already know Gauss’s guess was true, but thought primes were random, this would be a good reason to start believing Gauss. Since we already know Gauss’s guess is true, perhaps this is a good reason to think the primes are random!

Primes are random, take 2

Is there a pattern in the prime Remember the prime number theorem (Gauss’s guess): The numbers? count of primes up to x is well-approximated by x/ ln x, in the Paul Pollack sense that the relative error tends to 0. Intro

A mystery?

So what is a pattern anyway?

Randomness: A new hope

La ﬁn

68 / 74 Paul Pollack Is there a pattern in the prime numbers? If we didn’t already know Gauss’s guess was true, but thought primes were random, this would be a good reason to start believing Gauss. Since we already know Gauss’s guess is true, perhaps this is a good reason to think the primes are random!

Primes are random, take 2

A mystery? Theorem (Wunderlich)

So what is a pattern The probability that Gauss’s guess about π(x) is true for a anyway? random R-prime sequence is 100%. Randomness: A new hope

La ﬁn

69 / 74 Paul Pollack Is there a pattern in the prime numbers? Primes are random, take 2

A mystery? Theorem (Wunderlich)

So what is a pattern The probability that Gauss’s guess about π(x) is true for a anyway? random R-prime sequence is 100%. Randomness: A new hope La ﬁn If we didn’t already know Gauss’s guess was true, but thought primes were random, this would be a good reason to start believing Gauss. Since we already know Gauss’s guess is true, perhaps this is a good reason to think the primes are random! 70 / 74 Paul Pollack Is there a pattern in the prime numbers? It’s almost impossible to distinguish π(x) and li(x) on this graph. How close are they?

How to make a million bucks, the hard way

Is there a pattern in the Let’s put back up the graph comparing π(x), x/ ln x, and li(x) prime numbers? for x up to 100000. Paul Pollack

Intro 8000 A mystery?

So what is a pattern 6000 anyway?

Randomness: 4000 A new hope p X La ﬁn 2000 X logH XL li X 20 000 40 000 60 000 80 000 100 000 H L

H L

71 / 74 Paul Pollack Is there a pattern in the prime numbers? How to make a million bucks, the hard way

Is there a pattern in the Let’s put back up the graph comparing π(x), x/ ln x, and li(x) prime numbers? for x up to 100000. Paul Pollack

Intro 8000 A mystery?

So what is a pattern 6000 anyway?

Randomness: 4000 A new hope p X La ﬁn 2000 X logH XL li X 20 000 40 000 60 000 80 000 100 000 H L

It’s almost impossible to distinguish π(x) and li(x) onH L this

72 / 74 graph. How close are they? Paul Pollack Is there a pattern in the prime numbers? How to make a million bucks, ctd.

Is there a pattern in the prime numbers? The prime number theoem says that the relative error is small Paul Pollack (heading to zero), but what about the absolute error? Intro

A mystery? Conjecture (Riemann Hypothesis)

So what is a pattern For all x > 3, we have anyway? √ Randomness: |π(x) − li(x)| < x ln x. A new hope

La ﬁn The Clay Mathematics Institute oﬀers one million dollars if you can prove this. What are you waiting for?

73 / 74 Paul Pollack Is there a pattern in the prime numbers? Is there a pattern in the prime numbers?

Paul Pollack

Intro

A mystery?

So what is a pattern anyway? Thank you!

Randomness: A new hope

La ﬁn

74 / 74 Paul Pollack Is there a pattern in the prime numbers?