Exploring great mysteries about prime numbers - Wayzata Public ...

Exploring great mysteries about prime numbers BEN BRUBAKER (University of Minnesota – Twin Cities) [email protected]

Young Scientist Roundtable January 8, 2013

Ben Brubaker (UMN)

Mysteries of the primes

January 8, 2013

1 / 31

Big Questions for the next hour

What is mathematics? What kinds of questions do mathematicians try to solve? What do mathematicians do all day? How does mathematical proof resemble a poem or a painting? Why should we learn mathematics? How can we become better at mathematics?

Ben Brubaker (UMN)


January 8, 2013

2 / 31

A Mistaken Impression

Ben Brubaker (UMN)


January 8, 2013

3 / 31

Another Mistaken Impression

(source: sparknotes.com) Ben Brubaker (UMN)


January 8, 2013

4 / 31

What is Mathematics? “A mathematician, like a painter or a poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas.” – G. H. Hardy, A Mathematician’s Apology (1941) “...there is nothing as dreamy and poetic, nothing as radical, subversive, and psychedelic, as mathematics. It is every bit as mind blowing as cosmology or physics (mathematicians conceived of black holes long before astronomers actually found any), and allows more freedom of expression than poetry, art, or music (which depend heavily on properties of the physical universe). Mathematics is the purest of the arts, as well as the most misunderstood.” – Paul Lockhart, A Mathematician’s Lament (1987) Ben Brubaker (UMN)


January 8, 2013

5 / 31

A First Example from Geometry Here’s an example discussed in Lockhart’s “Lament”: Draw a rectangle and then a triangle contained within the rectangle as follows:

How much of the box does the area of the triangle take up?

Ben Brubaker (UMN)


January 8, 2013

6 / 31

A First Example from Geometry Here’s an example discussed in Lockhart’s “Lament”: Draw a rectangle and then a triangle contained within the rectangle as follows:

How much of the box does the area of the triangle take up? We can at least agree that, without prior knowledge, the answer requires some justification.... some additional argument.

Ben Brubaker (UMN)


January 8, 2013

6 / 31

Some Geometric Inspiration The next step is almost magic – we are inspired to draw a single (dotted) line:

Suddenly we see that the dotted line has chopped our rectangle into two pieces, and each piece is cut diagonally in half by the sides of the triangle. Where did the inspiration come from? Trial-and-error? Dumb luck? Lots of work with triangles and rectangles? That’s the magic... Now try to go further! Ben Brubaker (UMN)


January 8, 2013

7 / 31

Further games with triangles and rectangles Does this work for every triangle inscribed in a rectangle? What about...

The area of this triangle is less than half that of the rectangle.

Ben Brubaker (UMN)


January 8, 2013

8 / 31

Further games with triangles and rectangles Does this work for every triangle inscribed in a rectangle? What about...

The area of this triangle is less than half that of the rectangle. One fix:

Ben Brubaker (UMN)


January 8, 2013

8 / 31

A More Involved Example

In what follows, we repeat all of the key steps from the previous example: Ask a natural question Search for a spark of insight leading to a proof Marvel at the beauty, hunt for loose ends, broaden the scope Ask a further question, repeating the steps over again But now we investigate an entirely different subject of mathematics – number theory – and investigate much more deeply.

Ben Brubaker (UMN)


January 8, 2013

9 / 31

Prime numbers Number theory studies questions about whole numbers. The building blocks of whole numbers are the “prime numbers” – numbers whose only divisors are 1 and itself. Here’s a list of some prime numbers:

Figure: Euclid

2, 3, 5, 7, 11, 13, 17, . . . , 997, . . .

Fundamental Theorem of Arithmetic (Euclid, ∼300 BC) Every positive whole number may be factored uniquely as a product of primes (up to ordering).

Ben Brubaker (UMN)


January 8, 2013

10 / 31

Fundamental Theorem of Arithmetic (Euclid, ∼300 BC) Every positive whole number may be factored uniquely as a product of primes (up to ordering). For example, 147 = 3 × 72 (“Up to ordering” means this factorization is considered the same as writing 147 = 7 × 3 × 7, etc.) The fundamental theorem explains why we don’t consider the number 1 to be prime. If 1 were considered prime, the above result would be false: 21 = 3 × 7 = 1 × 3 × 7 The statement seems so obvious we take it for granted. But it requires proof. Perhaps we’ll discuss this in the session following refreshments. Ben Brubaker (UMN)


January 8, 2013

11 / 31

Euclid’s other theorem about primes How many prime numbers are there? Only finitely many? Maybe there are only 348 prime numbers. Or can we always find a prime larger than a given number?

Ben Brubaker (UMN)


January 8, 2013

12 / 31


Theorem (Euclid, ∼300 BC) There are infinitely many prime numbers.

Ben Brubaker (UMN)


January 8, 2013

12 / 31


Theorem (Euclid, ∼300 BC) There are infinitely many prime numbers. Proof: Suppose the conclusion of the theorem is false (i.e., that there are only finitely many prime numbers). Show how this leads to a logical impossibility, and hence the theorem must be true. In Latin, “reductio ad absurdum”

Ben Brubaker (UMN)


January 8, 2013

12 / 31

Euclid’s proof (continued) Theorem (Euclid, ∼300 BC) There are infinitely many prime numbers. Proof: Suppose there are only finitely many primes. In fact, suppose the only primes we knew were 2, 3, and 5. How could we construct a new prime using these?

Ben Brubaker (UMN)


January 8, 2013

13 / 31

Euclid’s proof (continued) Theorem (Euclid, ∼300 BC) There are infinitely many prime numbers. Proof: Suppose there are only finitely many primes. In fact, suppose the only primes we knew were 2, 3, and 5. How could we construct a new prime using these? Euclid’s elegant idea: Consider 2 · 3 · 5 + 1 = 31 We’ve made a new prime. This same construction works no matter how many primes we start with!

Ben Brubaker (UMN)


January 8, 2013

13 / 31

Euclid’s proof (continued) Theorem (Euclid, ∼300 BC) There are infinitely many prime numbers. Proof: Suppose there are only finitely many primes. In fact, suppose the only primes we knew were 2, 3, and 5. How could we construct a new prime using these? Euclid’s elegant idea: Consider 2 · 3 · 5 + 1 = 31 We’ve made a new prime. This same construction works no matter how many primes we start with! We don’t know the result is necessarily prime, but we do know that it is a number not divisible by any of our primes, so by unique factorization, must factor into NEW primes. (E.g., start with 2 & 7 : 2 × 7 + 1 = 15 = 3 × 5) Ben Brubaker (UMN)


January 8, 2013

13 / 31

Euclid’s proof (in full generality) Theorem (Euclid, ∼300 BC) There are infinitely many prime numbers. Proof: Suppose there are only finitely many primes. Say k of them. Then we can number them in increasing order: p1 (= 2), p2 (= 3), p3 (= 5), . . . , pk Consider the integer N = p1 · p2 · p3 · · · pk + 1. By unique factorization, it is expressible as a unique product of primes. But none of the primes p1 , . . . , pk can be factors. The number N, when divided by any of these primes, has remainder 1. So this finite list of {p1 , . . . , pk } can’t be all the primes. Contradiction! Ben Brubaker (UMN)


January 8, 2013

14 / 31

What are the prime numbers? Euler’s theorem tells us the sequence of prime numbers goes on forever, but doesn’t tell us which integers are prime. There’s no efficient rule or pattern for enumerating the prime numbers. For example, knowing that 997 is prime doesn’t help me to find the next larger prime. (Turns out it is 1009...)

Ben Brubaker (UMN)


January 8, 2013

15 / 31

What are the prime numbers? Euler’s theorem tells us the sequence of prime numbers goes on forever, but doesn’t tell us which integers are prime. There’s no efficient rule or pattern for enumerating the prime numbers. For example, knowing that 997 is prime doesn’t help me to find the next larger prime. (Turns out it is 1009...) Compare this to another famous sequence – the Fibonacci sequence: {1, 1, 2, 3, 5, 8, 13, 21, . . .} Then the nth Fibonacci number Fn is given by the formula " √ !n √ !n # 1 1+ 5 1− 5 Fn = √ − 2 2 5 Ben Brubaker (UMN)


January 8, 2013

15 / 31

A machine for generating (some) primes? Consider the value of 2p − 1 where p is prime: p 2 3 5 7 13 17 19 23

Ben Brubaker (UMN)

2p − 1 3 7 31 127 8191 131071 524287 8388607


January 8, 2013

16 / 31

A machine for generating (some) primes? Consider the value of 2p − 1 where p is prime: p 2 3 5 7 13 17 19 23

2p − 1 3 7 31 127 8191 131071 524287 8388607

All in this second column are prime except the last: 8388607 = 47 × 178481

Ben Brubaker (UMN)


January 8, 2013

16 / 31

Mersenne primes Primes of the form 2p − 1 with p prime are called Mersenne primes, after the 17th century French monk who tabulated many of them (with a few errors). The largest known prime is 243,112,609 − 1, a Mersenne prime. Today, such primes are found using shared processing power via the internet, e.g. G.I.M.P.S. Figure: Marin Mersenne

Proposition (Elementary) Suppose that ab − 1 is prime for some choice of integers a and b > 1. Then a = 2 and b is prime. That is, if ab − 1 is prime, it must be a Mersenne prime.

Ben Brubaker (UMN)


January 8, 2013

17 / 31

Generalizing Euclid’s result on the infinitude of primes We can ask for a proof that there are infinitely many primes in “thinner” sets than the integers. For example, are there infinitely many primes in the list: 1, 4, 7, 10, 13, 16, 19, . . .

Ben Brubaker (UMN)

(1 more than a multiple of 3 – (3n + 1))?


January 8, 2013

18 / 31



Yes! (Proof due to Dirichlet in 1837, using ingenious methods related to infinite series) And are there infinitely many primes in the list: 2, 5, 10, 17, 26, 37, 50, . . .

Ben Brubaker (UMN)

(1 more than a perfect square – (n2 + 1))?


January 8, 2013

18 / 31



Yes! (Proof due to Dirichlet in 1837, using ingenious methods related to infinite series) And are there infinitely many primes in the list: 2, 5, 10, 17, 26, 37, 50, . . .

(1 more than a perfect square – (n2 + 1))?

Unsolved, and extremely hard!

Ben Brubaker (UMN)


January 8, 2013

18 / 31

How fast does the number of primes grow?

Let π(x) denote the number of primes less than or equal to x. So π(10) = 4 since 2, 3, 5, and 7 are the 4 primes less than 10. In particular, π(x) < x since not every number less than x is prime. Is there a simple function that explains how π(x) grows as x gets larger and larger? Maybe it grows like 12 x? √ Or maybe like x? Or something even more magical?

Ben Brubaker (UMN)


January 8, 2013

19 / 31

Data on π(x) We want to know what π(x) looks like for x really big: x 101 104 108 1012 1016 1020 1024

Ben Brubaker (UMN)

π(x) 4 1.22 × 103 5.76 × 106 3.76 × 1010 2.79 × 1014 2.22 × 1018 1.84 × 1022


January 8, 2013

20 / 31


Ben Brubaker (UMN)

π(x) 4 1.22 × 103 5.76 × 106 3.76 × 1010 2.79 × 1014 2.22 × 1018 1.84 × 1022

% prime 40 12.2 5.76 3.76 2.79 2.22 1.84


x/π(x) 2.5 8.13 17.3 26.5 35.8 45.0 54.2

January 8, 2013

20 / 31


π(x) 4 1.22 × 103 5.76 × 106 3.76 × 1010 2.79 × 1014 2.22 × 1018 1.84 × 1022

% prime 40 12.2 5.76 3.76 2.79 2.22 1.84

x/π(x) 2.5 8.13 17.3 26.5 35.8 45.0 54.2

Even though the percentage looks to be dropping to 0, the number of digits in π(x) is almost keeping pace with the number of digits of x. √ √ So x is a bad guess. For example 1024 = 1012 . Our guess for π(x) should be really close to x. Ben Brubaker (UMN)


January 8, 2013

20 / 31

Making a conjecture about the growth of π(x) x 101 104 108 1012 1016 1020 1024

x/π(x) 2.5 8.13 17.3 26.5 35.8 45.0 54.2

What function inputs a number x and outputs the number of digits of x?

Ben Brubaker (UMN)


January 8, 2013

21 / 31

Making a conjecture about the growth of π(x) x 101 104 108 1012 1016 1020 1024

x/π(x) 2.5 8.13 17.3 26.5 35.8 45.0 54.2

What function inputs a number x and outputs the number of digits of x? This is the definition of the logarithm function (in base 10): log10 (1024 ) = 24 Not bad – it is the right order of magnitude – but we wanted 54.2. Ben Brubaker (UMN)


January 8, 2013

21 / 31

Making a conjecture about the growth of π(x) Idea: Change the base b of the logarithm! Remember the property of logs: logb (1024 ) =

log10 (1024 ) log10 (b)

So if we want b so that logb (1024 ) = 54.2, we substitute: 54.2 = 24/log10 (b) and rearrange terms: log10 (b) = 24/54.2 Then take both sides as exponents of 10: b = 1024/54.2 ≈ 2.77

Ben Brubaker (UMN)


January 8, 2013

22 / 31

Making a conjecture about the growth of π(x) Idea: Change the base b of the logarithm! Remember the property of logs: logb (1024 ) =

log10 (1024 ) log10 (b)

So if we want b so that logb (1024 ) = 54.2, we substitute: 54.2 = 24/log10 (b) and rearrange terms: log10 (b) = 24/54.2 Then take both sides as exponents of 10: b = 1024/54.2 ≈ 2.77 (In fact, further data would show that we should take b = 2.7182818... = e. That is, use “natural logarithm” ln(x)!) Ben Brubaker (UMN)


January 8, 2013

22 / 31

Remember π(x) is the number of primes less than or equal to x.

Prime Number Theorem (Hadamard, de la Vall´ee Poussin, 1896) As x gets larger and larger, π(x) gets closer and closer to x/ ln x. More precisely, π(x) = 1. x→∞ x/ ln x lim

The proof is spectacular, but way too involved to describe here. The standard proof uses something known as the “Riemann zeta function” which is manipulated using√calculus for the complex numbers (numbers of the form a + bi where i = −1.)

Ben Brubaker (UMN)


January 8, 2013

23 / 31

Big Questions Revisited What is mathematics? What kinds of questions do mathematicians try to solve? What do mathematicians do all day? How does mathematical proof resemble a poem or a painting? Why should we learn mathematics? How can we become better at mathematics?

Ben Brubaker (UMN)


January 8, 2013

24 / 31

Big Questions Revisited What is mathematics? What kinds of questions do mathematicians try to solve? What do mathematicians do all day? How does mathematical proof resemble a poem or a painting? Why should we learn mathematics? How can we become better at mathematics?

Thanks for your attention! Ben Brubaker (UMN)


January 8, 2013

24 / 31

Back to Euclid – How to factor numbers? Euclid’s theorem tells us that every number can be factored into a product of primes, but doesn’t give us an algorithm for finding these primes. Take a number like: 12345678910111213 What are its prime factors? Maybe it is prime itself?

Ben Brubaker (UMN)


January 8, 2013

25 / 31

Back to Euclid – How to factor numbers? Euclid’s theorem tells us that every number can be factored into a product of primes, but doesn’t give us an algorithm for finding these primes. Take a number like: 12345678910111213 What are its prime factors? Maybe it is prime itself? There are clever tests to check if a number is prime. This is not prime. But then what? Check small prime factors? Using a Computer Algebra System, we find 12345678910111213 = 113 × 125693 × 869211457

Ben Brubaker (UMN)


January 8, 2013

25 / 31

Factoring and cryptography

Because factoring large numbers is hard (but multiplying two numbers is easy), we can use it to build a system for encrypting and decrypting secret messages. It really does work like a lock. Anyone can put an open lock on a box (like multiplying large numbers), but it is very hard to undo it. You have to have a very precise key (like a prime factor). We’ll do an example of a “public key cryptosystem” developed around 1977 by Rivest, Shamir, and Adleman (known as the RSA cryptosystem).

Ben Brubaker (UMN)


January 8, 2013

26 / 31

A much too simple example of RSA Locking the box: 1

Pick two big primes p and q and multiply them. Call the result n. (For us, let p = 7 and q = 13 so that n = pq = 91).

2

Pick a number k less than n, and relatively prime to (p − 1) · (q − 1) [Remember, “relatively prime” means that they have no common divisors other than 1] (For us, k must be relatively prime to 6 · 12 = 72. Pick k = 11.)

3

Tell the world your two numbers n and k. Your “public key”

4

People send you messages (i.e. numbers a less than n) by finding the remainder of ak after dividing by n. (For example, to send a = 5, we compute the remainder of 511 /91. The answer is 73. People send us their encrypted message “73”. Key Point: Hard to find a knowing only the remainder of a11 /91) Ben Brubaker (UMN)


January 8, 2013

27 / 31

A much too simple example of RSA (cont.) Unlocking the box: 1

We are sent an encrypted message “73” using our public key (n, k). (For us, n = 91, k = 11)

2

Because we can factor n (but no one else can), we can compute this very important number (p − 1) · (q − 1). Call it φ(n). (For us, p − 1 = 6 and q − 1 = 12 so φ(n) = (p − 1) · (q − 1) = 72.)

3

Fermat’s Little Thm: For any integer a, the remainder of aφ(n) /n is 1! (For us, this means for any a, a72 /91 has remainder 1.) We can use this fact to find a, knowing the remainder of ak /n.

Ben Brubaker (UMN)


January 8, 2013

28 / 31

Finishing our example We are trying to determine a, knowing a11 /91 has remainder 73. Our advantage is that we can compute φ(91) = 72 and by Fermat’s Little Theorem: x 72 /91

always has remainder 1 for any integer x

Claim: We need only compute the remainder of 7359 /91. Why? Because 7359 and (a11 )59 have the same remainder upon dividing by 91. But... (a11 )59 = a649 = a

(law of multiplying exponents)

1+72·9

= a · (a9 )72

What happens when we divide by 91 and take remainders? Ben Brubaker (UMN)


January 8, 2013

29 / 31

Finishing our example We are trying to determine a, knowing a11 /91 has remainder 73. Our advantage is that we can compute φ(91) = 72 and by Fermat’s Little Theorem: x 72 /91

always has remainder 1 for any integer x

Claim: We need only compute the remainder of 7359 /91. Why? Because 7359 and (a11 )59 have the same remainder upon dividing by 91. But... (a11 )59 = a649 = a

(law of multiplying exponents)

1+72·9

= a · (a9 )72

What happens when we divide by 91 and take remainders? We get a! Ben Brubaker (UMN)


January 8, 2013

29 / 31

Summarizing key steps in RSA We can compute φ(n) = (p − 1) · (q − 1), but no one else can. Fermat’s little theorem gives us a way to solve for the remainder of ak /n for any a less than n if we know φ(n). FACT: We can always find a magic integer m so that (k · m)/φ(n) has remainder 1. This required k to be relatively prime to φ(n). For us, m = 59 so that 11 · 59 = 1 + 9 ∗ 72 and so dividing by 72 gives remainder 1. The method to do this is very fast, and uses the “Euclidean algorithm.” Euclid again! Ben Brubaker (UMN)


January 8, 2013

30 / 31

A real world example from RSA The following large composite, made from multiplying two primes, has not been factored:

RSA-220 = 226013852620340578494165404861019751350803 891571977671832119776810944564181796667660 859312130658257725063156288667697044807000 1811149711863002112487928199487482066070131 0665866460833279828035603792053919801399464 96955261 For a great reference on some of the basics we’ve covered, including Fermat’s Little Theorem, see: “A Friendly Introduction to Number Theory,” by Joseph H. Silverman Ben Brubaker (UMN)


January 8, 2013

31 / 31