Download Exploring great mysteries about prime numbers

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)
Mysteries of the primes
January 8, 2013
2 / 31
A Mistaken Impression
Ben Brubaker (UMN)
Mysteries of the primes
January 8, 2013
3 / 31
Another Mistaken Impression
(source: sparknotes.com)
Ben Brubaker (UMN)
Mysteries of the primes
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)
Mysteries of the primes
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)
Mysteries of the primes
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)
Mysteries of the primes
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)
Mysteries of the primes
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)
Mysteries of the primes
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)
Mysteries of the primes
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)
Mysteries of the primes
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)
Mysteries of the primes
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)
Mysteries of the primes
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)
Mysteries of the primes
January 8, 2013
12 / 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?
Theorem (Euclid, ∼300 BC)
There are infinitely many prime numbers.
Ben Brubaker (UMN)
Mysteries of the primes
January 8, 2013
12 / 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?
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)
Mysteries of the primes
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)
Mysteries of the primes
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)
Mysteries of the primes
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)
Mysteries of the primes
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)
Mysteries of the primes
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)
Mysteries of the primes
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)
Mysteries of the primes
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
Mysteries of the primes
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)
Mysteries of the primes
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)
Mysteries of the primes
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))?
Mysteries of the primes
January 8, 2013
18 / 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, . . .
(1 more than a multiple of 3 – (3n + 1))?
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))?
Mysteries of the primes
January 8, 2013
18 / 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, . . .
(1 more than a multiple of 3 – (3n + 1))?
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)
Mysteries of the primes
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)
Mysteries of the primes
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
Mysteries of the primes
January 8, 2013
20 / 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
% prime
40
12.2
5.76
3.76
2.79
2.22
1.84
Mysteries of the primes
x/π(x)
2.5
8.13
17.3
26.5
35.8
45.0
54.2
January 8, 2013
20 / 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
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)
Mysteries of the primes
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)
Mysteries of the primes
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)
Mysteries of the primes
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)
Mysteries of the primes
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)
Mysteries of the primes
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 Vallée 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)
Mysteries of the primes
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)
Mysteries of the primes
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)
Mysteries of the primes
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)
Mysteries of the primes
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)
Mysteries of the primes
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)
Mysteries of the primes
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)
Mysteries of the primes
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)
Mysteries of the primes
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)
Mysteries of the primes
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)
Mysteries of the primes
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)
Mysteries of the primes
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)
Mysteries of the primes
January 8, 2013
31 / 31