What we (don’t) know about primes
Paul-Olivier Dehaye
pdehaye@math.ethz.ch
Maturandentage, ETH Zürich
02 September 2009
The School of Athens
Rafael
The School of Athens
Rafael
Definition
A positive integer p > 1 is a prime number if it is not possible to
write it in the form
p = a × b,
where a > 1 and b > 1.
Definition
A positive integer p > 1 is a prime number if it is not possible to
write it in the form
p = a × b,
where a > 1 and b > 1. A number greater than 1 that is not prime
is called composite.
Definition
A positive integer p > 1 is a prime number if it is not possible to
write it in the form
p = a × b,
where a > 1 and b > 1. A number greater than 1 that is not prime
is called composite.
The numbers 6 (= 3 × 2), 51 (= 3 × 17), 2009 (= 49 × 41) are
composite.
Definition
A positive integer p > 1 is a prime number if it is not possible to
write it in the form
p = a × b,
where a > 1 and b > 1. A number greater than 1 that is not prime
is called composite.
The numbers 6 (= 3 × 2), 51 (= 3 × 17), 2009 (= 49 × 41) are
composite.
The numbers 2, 3, 5, 7, 11, 13, · · · , 2003,
2011, 2017, 2027, · · · , 243,112,609 − 1, · · ·
are prime.
Fundamental theorem of arithmetic
Euclid (300BC) proved that
A number can be factored in a
unique way as a product of primes.
Fundamental theorem of arithmetic
Euclid (300BC) proved that
A number can be factored in a
unique way as a product of primes.
For instance,
2009 = 72 × 41,
1 000 000 = 26 × 56 ,
7=7
Fundamental theorem of arithmetic
Euclid (300BC) proved that
A number can be factored in a
unique way as a product of primes.
For instance,
2009 = 72 × 41,
1 000 000 = 26 × 56 ,
7=7
This is the reason why we exclude 1 (and -1) as a prime number.
Otherwise, we would have
35 = 5 × 7 = 15 × 5 × 7 = (−1)2 × 5 × 7
which we don’t want to consider as different factorizations of 35.
There are infinitely many primes
Euclid also proved that
There are infinitely many primes.
There are infinitely many primes
Euclid also proved that
There are infinitely many primes.
His proof works by contradiction, and is one
of the most beautiful proofs in mathematics.
There are infinitely many primes
Euclid also proved that
There are infinitely many primes.
His proof works by contradiction, and is one
of the most beautiful proofs in mathematics.
The proof is by reductio ad absurdum, and reductio ad absurdum,
which Euclid loved so much, is
one of a mathematician’s favourite
weapons. It is a far finer gambit
than any chess gambit: a chess
player may offer the sacrifice of a
pawn or even a piece, but a mathematician offers the game.
There are infinitely many primes
Assume, for the sake of contradiction, that there are only k prime
numbers, and call them p1 , p2 , · · · , pk .
There are infinitely many primes
Assume, for the sake of contradiction, that there are only k prime
numbers, and call them p1 , p2 , · · · , pk .
Now, compute the number W = p1 p2 · · · pk + 1.
There are infinitely many primes
Assume, for the sake of contradiction, that there are only k prime
numbers, and call them p1 , p2 , · · · , pk .
Now, compute the number W = p1 p2 · · · pk + 1.
Is this number prime? It cannot be, since we have that W > pi for
all i.
There are infinitely many primes
Assume, for the sake of contradiction, that there are only k prime
numbers, and call them p1 , p2 , · · · , pk .
Now, compute the number W = p1 p2 · · · pk + 1.
Is this number prime? It cannot be, since we have that W > pi for
all i.
So W has to be composite, and hence W can be expressed as a
product of primes. In particular, W is divisible by at least one
prime, say pr .
There are infinitely many primes
Assume, for the sake of contradiction, that there are only k prime
numbers, and call them p1 , p2 , · · · , pk .
Now, compute the number W = p1 p2 · · · pk + 1.
Is this number prime? It cannot be, since we have that W > pi for
all i.
So W has to be composite, and hence W can be expressed as a
product of primes. In particular, W is divisible by at least one
prime, say pr .
This is a contradiction, because pr cannot divide both W and
W − 1 = p1 · · · pk .
There are infinitely many primes
Assume, for the sake of contradiction, that there are only k prime
numbers, and call them p1 , p2 , · · · , pk .
Now, compute the number W = p1 p2 · · · pk + 1.
Is this number prime? It cannot be, since we have that W > pi for
all i.
So W has to be composite, and hence W can be expressed as a
product of primes. In particular, W is divisible by at least one
prime, say pr .
This is a contradiction, because pr cannot divide both W and
W − 1 = p1 · · · pk .
We can therefore conclude that our assumption that there exist
only k primes was wrong.
Building primes? Hard!
Fermat thought that a number of the form Fn =
n
22 + 1 is always prime.
For instance, he knew that
0
1
2
F0 = 3 = 22 +1, F1 = 5 = 22 +1, F2 = 17 = 22 +1,
3
4
F3 = 257 = 22 + 1, F4 = 65537 = 22 + 1
are prime.
Building primes? Hard!
Fermat thought that a number of the form Fn =
n
22 + 1 is always prime.
For instance, he knew that
0
1
2
F0 = 3 = 22 +1, F1 = 5 = 22 +1, F2 = 17 = 22 +1,
3
4
F3 = 257 = 22 + 1, F4 = 65537 = 22 + 1
are prime. Unfortunately he didn’t know that
F5 = 232 + 1 = 641 × 6 700 417.
Building primes? Hard!
Fermat thought that a number of the form Fn =
n
22 + 1 is always prime.
For instance, he knew that
0
1
2
F0 = 3 = 22 +1, F1 = 5 = 22 +1, F2 = 17 = 22 +1,
3
4
F3 = 257 = 22 + 1, F4 = 65537 = 22 + 1
are prime. Unfortunately he didn’t know that
F5 = 232 + 1 = 641 × 6 700 417.
A Mersenne prime is a prime number of the
form 2p − 1, where p itself is a prime.
For instance, 7 = 23 − 1, 25 − 1, 27 − 1,
243 112 609 − 1, but not 211 − 1.
Patterns of primes
Twin primes are pairs of primes p, p + 2. We do not know if there
are infinitely many.
Patterns of primes
Twin primes are pairs of primes p, p + 2. We do not know if there
are infinitely many.
Observe that p, p + 2 × 1 and p + 2 × 2 can never all be prime at
once.
Patterns of primes
Twin primes are pairs of primes p, p + 2. We do not know if there
are infinitely many.
Observe that p, p + 2 × 1 and p + 2 × 2 can never all be prime at
once. However, 3, 3 + 4 × 1 = 7 and 3 + 4 × 2 = 11 are. Also, all of
6 171 054 912 832 631 + 81 737 658 082 080 × n
for n = 0 to 24 are prime (2008).
Patterns of primes
Twin primes are pairs of primes p, p + 2. We do not know if there
are infinitely many.
Observe that p, p + 2 × 1 and p + 2 × 2 can never all be prime at
once. However, 3, 3 + 4 × 1 = 7 and 3 + 4 × 2 = 11 are. Also, all of
6 171 054 912 832 631 + 81 737 658 082 080 × n
for n = 0 to 24 are prime (2008).
There exist arbitrarly long arithmetic progressions of primes
(proved in 2004).
Testing for primes
The problem of testing for primes is relatively easy, but this is a
recent result. In 2002, Agrawal, Kayal and Saxena found an
algorithm that determines whether a number n is prime or not in
time (bounded by a) polynomial in the number of digits of n.
Testing for primes
The problem of testing for primes is relatively easy, but this is a
recent result. In 2002, Agrawal, Kayal and Saxena found an
algorithm that determines whether a number n is prime or not in
time (bounded by a) polynomial in the number of digits of n.
Still, there exists an algorithm that checks much faster if a number
of the form 2p − 1 is prime or not. That’s why Mersenne primes are
so popular.
Factoring? Hard!
Given the fundamental theorem of arithmetic, that says that any
number n factorizes in a unique way, it is natural to ask for an
algorithm to find this factorization.
Factoring? Hard!
Given the fundamental theorem of arithmetic, that says that any
number n factorizes in a unique way, it is natural to ask for an
algorithm to find this factorization.
In comparison to testing whether a number is prime, this is a much
more difficult exercise, which is (partly) a good thing.
Factoring? Hard!
Given the fundamental theorem of arithmetic, that says that any
number n factorizes in a unique way, it is natural to ask for an
algorithm to find this factorization.
In comparison to testing whether a number is prime, this is a much
more difficult exercise, which is (partly) a good thing. Much of
modern cryptography (hiding information) depends on this and
related questions.
Factoring? Hard!
Given the fundamental theorem of arithmetic, that says that any
number n factorizes in a unique way, it is natural to ask for an
algorithm to find this factorization.
In comparison to testing whether a number is prime, this is a much
more difficult exercise, which is (partly) a good thing. Much of
modern cryptography (hiding information) depends on this and
related questions. Primes are essential for many modern
applications: credit cards, cell phones, email, e-commerce, ...
Factoring? Hard!
Given the fundamental theorem of arithmetic, that says that any
number n factorizes in a unique way, it is natural to ask for an
algorithm to find this factorization.
In comparison to testing whether a number is prime, this is a much
more difficult exercise, which is (partly) a good thing. Much of
modern cryptography (hiding information) depends on this and
related questions. Primes are essential for many modern
applications: credit cards, cell phones, email, e-commerce, ...
For example, the Massey-Omura Three-Pass Protocol uses prime
numbers and allows for secure communication without the
exchange of keys (1982).
Three Pass Protocol (Shamir)
For A to send a message to B:
Three Pass Protocol (Shamir)
For A to send a message to B:
I
A encodes the message as a number m
Three Pass Protocol (Shamir)
For A to send a message to B:
I
A encodes the message as a number m
I
A picks a large prime p > m and
communicates it openly to B
Three Pass Protocol (Shamir)
For A to send a message to B:
I
A encodes the message as a number m
I
A picks a large prime p > m and
communicates it openly to B
I
A and B pick independently and
secretly the numbers eA and eB
Three Pass Protocol (Shamir)
For A to send a message to B:
I
A encodes the message as a number m
I
A picks a large prime p > m and
communicates it openly to B
I
A and B pick independently and
secretly the numbers eA and eB
I
A and B each compute the associated
numbers dA and dB such that
dA eA ≡ 1 mod p − 1 and dB eB ≡ 1
mod p − 1.
Three Pass Protocol (Shamir)
For A to send a message to B:
I
A encodes the message as a number m
I
A picks a large prime p > m and
communicates it openly to B
I
A and B pick independently and
secretly the numbers eA and eB
I
A and B each compute the associated
numbers dA and dB such that
dA eA ≡ 1 mod p − 1 and dB eB ≡ 1
mod p − 1.
Then, we have the essential fact (Fermat’s Little Theorem) that
x dA eA mod p ≡ x and x dB eB mod p ≡ x for any x.
Three Pass Protocol (Shamir)
For A to send a message to B:
I
A encodes the message as a number m
I
A picks a large prime p > m and
communicates it openly to B
I
A and B pick independently and
secretly the numbers eA and eB
I
A and B each compute the associated
numbers dA and dB such that
dA eA ≡ 1 mod p − 1 and dB eB ≡ 1
mod p − 1.
Then, we have the essential fact (Fermat’s Little Theorem) that
x dA eA mod p ≡ x and x dB eB mod p ≡ x for any x.
This would allow you to transmit a secret conversation across the
room, without having agreed beforehand on secret keys!
Density
Now back to what we know about primes... There are infinitely
many primes, but how frequent are they?
Density
Now back to what we know about primes... There are infinitely
many primes, but how frequent are they?
Define π(x) = the number of primes ≤ x.
Density
Now back to what we know about primes... There are infinitely
many primes, but how frequent are they?
Define π(x) = the number of primes ≤ x.
ΠHxL
æ æ
æ æ æ æ æ æ
15
æ æ æ æ æ æ
æ æ æ æ
æ æ
æ æ æ æ
æ æ æ æ æ æ
10
æ æ
æ æ æ æ æ æ
æ æ æ æ
æ æ
æ æ æ æ
5
æ æ
æ æ æ æ
æ æ
æ æ
æ
æ
10
20
30
40
50
60
n
Density
ΠHxL
150
100
50
200
400
600
800
n
1000
Density
Prime Number Theorem (1896)
Prime Number Theorem (1896)
Prime Number Theorem (1896)
Prime Number Theorem (1896)
When n is very large,
n
π(n) ∼
∼ Li(n) =
log n
Z
2
n
1
dx,
log x
which means the relative error between those functions goes to 0,
or that the chance of a large number n being prime tends to log1 n .
Prime Number Theorem (1896)
ΠHxL
LiHnL
ΠHxL
n
logHnL
150
100
50
200
400
600
800
1000
n
Prime Number Theorem (1896)
ΠHxL
LiHnL
ΠHxL
n
logHnL
150
100
50
200
400
600
800
1000
n
One could guess that π(x) ≤ Li(x). Littlewood
proved this is false, and it is thought that the first
reversal occurs around 1.397 × 10316 .
Prime Number Theorem for AP
One can ask the same question for arithmetic
progressions. Dirichlet proved the primes
distributed in a very natural way among
them.
Prime Number Theorem for AP
One can ask the same question for arithmetic
progressions. Dirichlet proved the primes
distributed in a very natural way among
them.
Birth of Analytic Number Theory
ζ(s) := 1
Birth of Analytic Number Theory
ζ(s) := 1 +
1
2s
Birth of Analytic Number Theory
ζ(s) := 1 +
1
1
+ s
s
2
3
Birth of Analytic Number Theory
ζ(s) := 1 +
1
1
1
+ s + s
s
2
3
4
Birth of Analytic Number Theory
ζ(s) := 1 +
1
1
1
1
1
1
+ s + s + s + s + s + ···
s
2
3
4
5
6
7
Birth of Analytic Number Theory
1
1
1
1
1
1
+ s + s + s + s + s + ···
s
2
3
4
5
6
7
1
1
1
1
1
1
+ s + s + s + s + s + ···
2s
4
6
8
10
12
ζ(s) := 1 +
1
ζ(s)
2s
=
Birth of Analytic Number Theory
1
1
1
1
+ s + s + s +
s
2
3
4
5
1
1
1
1
1
+ s + s + s + s
s
2
4
6
8
10
1
1
1
1
1+ s + s + s + s +
3
5
7
9
ζ(s) := 1 +
1−
1
2s
1
ζ(s)
2s
=
ζ(s)
=
1
1
+ s + ···
s
6
7
1
+ s + ···
12
1
1
+ s +·
11s
13
Birth of Analytic Number Theory
1
1
1
1
1
1
+ s + s + s + s + s + ···
s
2
3
4
5
6
7
1
1
1
1
1
1
+ s + s + s + s + s + ···
2s
4
6
8
10
12
1
1
1
1
1
1
1+ s + s + s + s + s + s +·
3
5
7
9
11
13
1
1
1
1
1
+ s + s + s + s + ···
s
3
9
15
21
27
ζ(s) := 1 +
1
ζ(s)
2s
1
ζ(s)
2s
1
1
1 − s ζ(s)
s
3
2
1−
=
=
=
Birth of Analytic Number Theory
1
1
1
1
1
1
+ s + s + s + s + s + ···
s
2
3
4
5
6
7
1
1
1
1
1
1
+ s + s + s + s + s + ···
2s
4
6
8
10
12
1
1
1
1
1
1
1+ s + s + s + s + s + s +·
3
5
7
9
11
13
1
1
1
1
1
+ s + s + s + s + ···
s
3
9
15
21
27
1
1
1
1
1
1 + s + s + s + s + s + ···
5
7
11
13
17
ζ(s) := 1 +
1
ζ(s)
2s
1
ζ(s)
2s
1
1
1 − s ζ(s)
s
3
2
1
1
1− s
1 − s ζ(s)
3
2
1−
=
=
=
=
Birth of Analytic Number Theory
1
1
1
1
1
1
+ s + s + s + s + s + ···
s
2
3
4
5
6
7
1
1
1
1
1
1
+ s + s + s + s + s + ···
2s
4
6
8
10
12
1
1
1
1
1
1
1+ s + s + s + s + s + s +·
3
5
7
9
11
13
1
1
1
1
1
+ s + s + s + s + ···
s
3
9
15
21
27
1
1
1
1
1
1 + s + s + s + s + s + ···
5
7
11
13
17
ζ(s) := 1 +
1
ζ(s)
2s
1
ζ(s)
2s
1
1
1 − s ζ(s)
s
3
2
1
1
1− s
1 − s ζ(s)
3
2
1−
=
=
=
=
Birth of Analytic Number Theory
1
1
1
1
1
1
+ s + s + s + s + s + ···
s
2
3
4
5
6
7
1
1
1
1
1
1
+ s + s + s + s + s + ···
2s
4
6
8
10
12
1
1
1
1
1
1
1+ s + s + s + s + s + s +·
3
5
7
9
11
13
1
1
1
1
1
+ s + s + s + s + ···
s
3
9
15
21
27
1
1
1
1
1
1 + s + s + s + s + s + ···
5
7
11
13
17
ζ(s) := 1 +
1
ζ(s)
2s
1
ζ(s)
2s
1
1
1 − s ζ(s)
s
3
2
1
1
1− s
1 − s ζ(s)
3
2
1−
=
=
=
=
1
1
1
1
··· 1 − s
1− s
1− s
1 − s ζ(s) = 1
7
5
3
2
Birth of Analytic Number Theory
1
1
1
1
1
1
+ s + s + s + s + s + ···
s
2
3
4
5
6
7
1
1
1
1
1
1
+ s + s + s + s + s + ···
2s
4
6
8
10
12
1
1
1
1
1
1
1+ s + s + s + s + s + s +·
3
5
7
9
11
13
1
1
1
1
1
+ s + s + s + s + ···
s
3
9
15
21
27
1
1
1
1
1
1 + s + s + s + s + s + ···
5
7
11
13
17
ζ(s) := 1 +
1
ζ(s)
2s
1
ζ(s)
2s
1
1
1 − s ζ(s)
s
3
2
1
1
1− s
1 − s ζ(s)
3
2
1−
=
=
=
=
1
1
1
1
··· 1 − s
1− s
1− s
1 − s ζ(s) = 1
7
5
3
2
ζ(s) =
X 1
Y 1
=
ns
1 − p1s
n
p
Plot of ζ
1.6
1.4
1.2
1.0
0.8
2
4
ζ(s) =
6
X 1
Y 1
=
s
n
1 − p1s
n
p
8
10
Plot of ζ
1.6
1.4
1.2
1.0
0.8
2
4
ζ(s) =
6
8
10
X 1
Y 1
=
s
n
1 − p1s
n
p
This gives a new proof that there are infinitely many primes!
Riemann (1859)
Riemann (1859)
ζ(s) =
X 1
Y 1
=
ns
1 − p1s
n
p
Riemann (1859)
ζ(s) =
X 1
Y 1
=
ns
1 − p1s
n
p
Riemann’s idea was to use complex
analysis: he put in complex numbers
instead of s.
Riemann (1859)
ζ(s) =
X 1
Y 1
=
ns
1 − p1s
n
p
Riemann’s idea was to use complex
analysis: he put in complex numbers
instead of s.
He started a whole new field in mathematics. It was very successful
at proving many results, such as the Prime Number Theorem
(density) or Dirichlet’s Theorem (arithmetic progressions), as well
as many others.
Riemann (1859)
ζ(s) =
X 1
Y 1
=
ns
1 − p1s
n
p
Riemann’s idea was to use complex
analysis: he put in complex numbers
instead of s.
He started a whole new field in mathematics. It was very successful
at proving many results, such as the Prime Number Theorem
(density) or Dirichlet’s Theorem (arithmetic progressions), as well
as many others.
However, this also led to a major open question in mathematics,
the Riemann Hypothesis.
First statement
ΠHxL
LiHnL
ΠHxL
n
logHnL
150
100
50
200
400
600
800
1000
n
First statement
ΠHxL
LiHnL
ΠHxL
n
logHnL
150
100
50
200
400
600
800
1000
n
The first statement of the Riemann Hypothesis, which is very
tangible, is that
1
|Li(n) − π(n)| ≤ C · n 2 +
Second statement
We now give a statement that is equivalent to the first one.
Second statement
We now give a statement that is equivalent to the first one.
The second statement requires analytic continuation and concerns
the zeros of the ζ function.
Second statement
We now give a statement that is equivalent to the first one.
The second statement requires analytic continuation and concerns
the zeros of the ζ function.
1.6
1.4
1.2
1.0
0.8
2
4
6
8
10
Second statement
The Riemann Hypothesis, in its original form, states that
If ζ(s) = 0 and <s > 0 then <s = 1/2.
Second statement
The Riemann Hypothesis, in its original form, states that
If ζ(s) = 0 and <s > 0 then <s = 1/2.
This has been verified for the first 1013 zeroes to be true...
Second statement
The Riemann Hypothesis, in its original form, states that
If ζ(s) = 0 and <s > 0 then <s = 1/2.
This has been verified for the first 1013 zeroes to be true...
but this is not a proof!
Conclusion
Conclusion
I
Pure math problem
Conclusion
I
Pure math problem
I
Real-world applications
Conclusion
I
Pure math problem
I
Real-world applications
I
Fascinating ideas
Conclusion
I
Pure math problem
I
Real-world applications
I
Fascinating ideas
I
Constant progress
Conclusion
I
Pure math problem
I
Real-world applications
I
Fascinating ideas
I
Constant progress
Thank you!