the 2003 pauli lectures - eth zürich - homepageeuler (1707-1783) euler’s father wanted his son to...

The 2003 Pauli LecturesENRICO BOMBIERI

IAS, PRINCETON, NJ

– p.1/45

THE LECTURES

Past, Present, Future

I. Arithmetic and Analysis: From Primes tothe Zeta Function

II. Arithmetic and Geometry: DiophantineEquations

III. The Rosetta Stone of -functions

– p.2/45

THE LECTURES

Past, Present, Future

I. Arithmetic and Analysis: From Primes tothe Zeta Function

II. Arithmetic and Geometry: DiophantineEquations

III. The Rosetta Stone of

�

-functions

– p.2/45

EARLY ARITHMETIC

Arithmetic(Andrea Bonaiuti, Church of S. Maria Novella, Florence)

– p.3/45

THE CAST OF CHARACTERS

CHINA, BRAHMEGUPTA, EUCLID

DIOPHANTUS, FIBONACCI

. . . . . .

FERMAT

– p.4/45

CHINA � 500 BC

The Chinese Little TheoremTheorem: If is prime then divides .

The Chinese Remainder Theorem (Gauss’s form)

Theorem: If with , , ...relatively prime in pairs and if

then is a solution of

– p.5/45

CHINA � 500 BC

The Chinese Little TheoremTheorem: If � is prime then � divides

� �

��

.


Theorem: If with , , ...relatively prime in pairs and if

then is a solution of

– p.5/45

CHINA � 500 BC

The Chinese Little TheoremTheorem: If � is prime then � divides

� �

��

.


Theorem: If � � �� with �� , �� , ...relatively prime in pairs and if

��

then � � �� is a solution of

� ��

– p.5/45

EUCLID � 300 BC

Theorem: The sequence of prime numbers cannotend.

Proof: Let be our list of knownprimes. The integer

has remainder when divided by any prime in thelist. Hence either it is a new prime or is a product ofnew primes. Thus we can always enlarge the list.QED

– p.6/45

EUCLID � 300 BC

Theorem: The sequence of prime numbers cannotend.

Proof: Let

�� be our list of knownprimes. The integer

� � � � � � ��

has remainder

�

when divided by any prime in thelist. Hence either it is a new prime or is a product ofnew primes. Thus we can always enlarge the list.QED

– p.6/45

VARIATION ON THEME

Proof: Every divisor of

� � ��

is either

�or greater

than � . Hence this number admits a prime factorlarger than � . QED

Exercise: Show that every prime can be obtained asa divisor of for a suitable .Hint: Try . (Wilson’s theorem, ascribedby E. Waring to Sir John Wilson (1741–1793),proved by Lagrange in 1773.)

Exercise: Show that is prime if and only ifdivides .

– p.7/45

VARIATION ON THEME


� � ��

is either

�or greater


Exercise: Show that every prime can be obtained asa divisor of � � �

for a suitable � .Hint: Try � � � �

�. (Wilson’s theorem, ascribed

by E. Waring to Sir John Wilson (1741–1793),proved by Lagrange in 1773.)

Exercise: Show that is prime if and only ifdivides .

– p.7/45

VARIATION ON THEME


� � ��

is either

�or greater


Exercise: Show that every prime can be obtained asa divisor of � � �

for a suitable � .Hint: Try � � � �

�. (Wilson’s theorem, ascribed

by E. Waring to Sir John Wilson (1741–1793),proved by Lagrange in 1773.)

Exercise: Show that � is prime if and only if �

divides�

� ��

.– p.7/45

FIBONACCI � 1200

Fibonacci

The Liber Quadratorum

Fibonacci’s formula

Congruent numbers

Fibonacci:

The congruent numberThe smallest solution is (Zagier)

,

.

– p.8/45

FIBONACCI � 1200

Fibonacci

Fibonacci sequence

��



Congruent numbers

Fibonacci:


,

.

– p.8/45

FIBONACCI � 1200

Fibonacci



� � � � � ��

Congruent numbers

Fibonacci:


,

.

– p.8/45

FIBONACCI � 1200

Fibonacci



� � � � � ��

Congruent numbers �

� � ��

� � � � � � � ��

Fibonacci: � � � �


,

.

– p.8/45

FIBONACCI � 1200

Fibonacci



� � � � � ��

Congruent numbers �

� � ��

� � � � � � � ��

Fibonacci: � � � �

The congruent number

� � �

The smallest solution is (Zagier)� �

� ��

�� ,

� ��

� � � � �� .– p.8/45

THE NEW CAST OF CHARACTERS

EULER, GAUSS, DIRICHLET

ABEL, JACOBI, GALOIS, RIEMANN

KUMMER, DEDEKIND, HILBERT

. . . . . .

– p.9/45

EULER (1707-1783)

– p.10/45

EULER (1707-1783)

Leonhard Eulerfounder of modern mathematics

– p.10/45

EULER (1707-1783)

Euler’s father wanted his son to follow him into the church and sent him to the

University of Basel to prepare for the ministry. He entered the University in 1720,

at the age of 14. He began his study of theology in the autumn of 1723 but,

although he was to be a devout Christian all his life, his enthousiasm was for

mathematics. He obtained his father’s consent to change to mathematics after

Johann Bernoulli had used his persuasion.

After Basel he went to St. Petersbourg in 1727, then back to Basel in 1733 where

he married Katharina Gsell and they had 13 children, but only five survived. Euler

claimed that he made some of his greatest mathematical discoveries while holding

a baby in his arms with other children playing round his feet. After a period in

Berlin he went back to St. Petersbourg in 1766, at the age of 59. Although he

became totally blind in 1771, half of his work was done during this second period

in St. Petersbourg.– p.10/45

EULER, PRIMES AND �

The most famous function in number theory:

It makes its first appearance in Euler’s famousequation

(In a letter to Daniel Bernoulli in 1736. Bernoulli found it “sehr

merkwürdig”)

– p.11/45

EULER, PRIMES AND �The most famous function in number theory:

� ��

� � �



merkwürdig”)

– p.11/45

EULER, PRIMES AND �The most famous function in number theory:

� ��

� � �


�

��

��

��

��

� ��


merkwürdig”)

– p.11/45

EULER AND , , ...

Euler formula forTheorem:

where are the Bernoulli numbers defined by

In particular, , ,

, .

Problem: What about

– p.12/45

EULER AND , , ...

Euler formula for

� � � �

Theorem:

� � � � � ��

� � � ��

� � � � ��

where � are the Bernoulli numbers defined by

��

�

��

� � � ��

In particular, , ,

, .

Problem: What about

– p.12/45

EULER AND , , ...

Euler formula for

� � � �

Theorem:

� � � � � ��

� � � ��

� � � � ��


��

�

��

� � � ��

In particular

� � � � � � �� ,

� �� ,

� � � � � � �� ,

� � � � � � � �� ,

� � � � � � � � � �

�� .

Problem: What about

– p.12/45

EULER AND , , ...

Euler formula for

� � � �

Theorem:

� � � � � ��

� � � ��

� � � � ��


��

�

��

� � � ��

In particular

� � � � � � �� ,

� �� ,

� � � � � � �� ,

� � � � � � � �� ,

� � � � � � � � � �

�� .

Problem: What about� � � � � � ��

��

��

��

�

– p.12/45

EULER AND

Euler tried hard to obtain a closed formula for� ��

, tono avail. In his 1785 Opuscula analytica he writes:

“Huiusmodi igitur casum singularem, qui talemrelationem non respuere videtur, hic accuratiusevolvam, scilicet summam seriei reciprocæ cubarum

, quam nullo adhucmodo sive ad circulum sive ad logarithmos reducerepotui.”

“At this point I will examine in rather more detail a unique case, which does

not seem alien to follow such a relation, namely the sum of the series of the

reciprocals of cubes , which so far

in no way I could reduce to the circle or to logarithms.”

– p.13/45

EULER AND




� ��

��

��

�� , quam nullo adhuc

modo sive ad circulum sive ad logarithmos reducerepotui.”



reciprocals of cubes , which so far


– p.13/45

EULER AND




� ��

��

��

�� , quam nullo adhuc

modo sive ad circulum sive ad logarithmos reducerepotui.”



reciprocals of cubes� � �� , which so far


– p.13/45

EULER AND , II

He continues by asking whether is a rationalcombination of and .

Euler also studied multiple zeta sums, such as

arising from his quest for a formula for .

– p.14/45

EULER AND , II

He continues by asking whether

� ��

is a rationalcombination of

��

and

�� .



– p.14/45

EULER AND , II


� ��


��

and

�� .

Motivation:

� � � �� appears in the expansions

��

� �

and

��

� � � ��

�� .



– p.14/45

EULER AND , II


� ��


��

and

�� .


� �

��

arising from his quest for a formula for

� ��

.

– p.14/45

EULER AND , II


� ��


��

and

�� .


� �

��


� ��

.

The problem of understanding relations amongmultiple zeta sums remains of great interest, linkedwith algebraic geometry (Deligne).

– p.14/45

EULER AND , II


� ��


��

and

�� .


� �

��


� ��

.

Theorem (Apery, 1979):

� ��

is irrational.

– p.14/45

EULER AND PRIMES

Euler’s Product FormulaIn the first volume of Euler’s book of 1748Introductio in Analysin Infinitorum we find theformula

.

Proof: A product of sums is the sum of all productswhose factors are obtained by taking one term fromeach sum. Expanding andmultiplying the sums we obtain the sum of allproducts . Euler’s identity follows fromunique factorization. QED

– p.15/45

EULER AND PRIMES


� � � � � � � � �

� � � �

� �

� � � � �

.

Proof: A product of sums is the sum of all productswhose factors are obtained by taking one term fromeach sum. Expanding andmultiplying the sums we obtain the sum of allproducts . Euler’s identity follows fromunique factorization. QED

– p.15/45

EULER AND PRIMES


� � � � � � � � �

� � � �

� �

� � � � �

.

Proof: A product of sums is the sum of all productswhose factors are obtained by taking one term fromeach sum. Expanding

� �

� � � � �

�

� � � �

andmultiplying the sums we obtain the sum of allproducts

� ��

. Euler’s identity follows fromunique factorization. QED

– p.15/45

TWO INTUITIONS BY EULER

Euler used divergent series freely (but correctly!):

Euler, 1737: .

“Proof”: Write , , etc. Then we havethe (formal) equality Now

and setting we have. Since is convergent,

one concludes by taking logarithms. QED

Euler, 1762/63: The number of primes up to is about .

This statement is usually attributed to the young Gauss, whowrote it in 1791 on the margin of the collection ofmathematical tables by J.C. Schulze (1778).

– p.16/45



Euler, 1737:

�� .

“Proof”: Write , , etc. Then we havethe (formal) equality Now

and setting we have. Since is convergent,

one concludes by taking logarithms. QED



– p.16/45



Euler, 1737:

�� .

“Proof”: Write

� � � �� ,

� � � �� , etc. Then we have

the (formal) equality � � � � � � �� Now

� � � � �� and setting � � �

we have

� � �� . Since

� �� is convergent,one concludes by taking logarithms. QED



– p.16/45



Euler, 1737:

�� .

“Proof”: Write

� � � �� ,

� � � �� , etc. Then we have

the (formal) equality � � � � � � �� Now

� � � � �� and setting � � �

we have

� � �� . Since

� �� is convergent,one concludes by taking logarithms. QED

Euler, 1762/63: The number of primes up to � is about � � � � � � .This statement is usually attributed to the young Gauss, whowrote it in 1791 on the margin of the collection ofmathematical tables by J.C. Schulze (1778).

– p.16/45

EULER’S FUNCTIONAL EQUATION

Euler method for studying functions defined by infinite series:Extend the domain of definition beyond the domain of convergence,using summability methods. His results for

� ��

��

�� :

Theorem: and

for .

Conjecture (1749):

– p.17/45



� ��

��

�� :

Theorem: � � � � � � �

and � ��

for

� � ��

�� .

Conjecture (1749):

– p.17/45



� ��

��

�� :

Theorem: � � � � � � �

and � ��

for

� � ��

�� .

Conjecture (1749):

� � � � � ��

� � � � � � � ��

��

� � � � � � ��

� � �

– p.17/45

MORE FUNCTIONAL EQUATIONS

Functional equations for ,(conjectured by Euler) and

were proved by Malmsténin 1849.

Functional equations, asymmetric form

Theorem: ,

.

Theorem (Eisenstein, 1849): For , in the open interval

and for all we have

– p.18/45


Functional equations for

� � ��

�� ,

� � ��

�� (conjectured by Euler) and

� � ��

��

��

��

�� were proved by Malmsténin 1849.


Theorem: ,

.


and for all we have

– p.18/45



� � ��

�� ,

� � ��


� � ��

��

��

��



Theorem:

� ��

� � � ��

,

� ��

��

� � � � � � ��

�� .


and for all we have

– p.18/45



� � ��

�� ,

� � ��


� � ��

��

��

��



Theorem:

� ��

� � � ��

,

� ��

��

� � � � � � ��

�� .

Theorem (Eisenstein, 1849): For � , �

in the open interval

� ��

� �

and for all � we have��

��

� �� !� " # �

� $&% � � � � � � �� !� � # � �� % � � '

– p.18/45

DIRICHLET (1805-1859)

– p.19/45


Dirichletfounder of analytic number theory

– p.19/45


Lejeune Dirichlet’s family came from the Belgian town of Richelet

where Dirichlet’s grandfather lived. This explains the origin of his

name which comes from "Le jeune de Richelet" meaning "The

young man from Richelet". His father was the postmaster of Düren,

the town of his birth situated about halfway between Aachen and

Cologne. Even before he entered the Gymnasium in Bonn in 1817,

at the age of 12, he had developed a passion for mathematics and

spent his pocket-money on buying mathematics books. At the

Gymnasium he was a model pupil.

He is recognized today for his fundamental contributions to number

theory and analysis.

– p.19/45

DIRICHLET, 1837

Introduced the Dirichlet -series

– p.20/45

DIRICHLET, 1837

Introduced the Dirichlet

�

-series

� ��

� ��

� � �

– p.20/45

DIRICHLET, 1837


�

-series

� ��

� ��

� � �

Here � ��

is a Dirichlet character

� � � ��

: Afunction on integers witha) � �� and � ��

if � ,� have a commonfactorb) multiplicativity � ��

c) periodicity � ��

– p.20/45

DIRICHLET, 1837


�

-series

� ��

� ��

� � �

Euler product:

� ��

– p.20/45

DIRICHLET, 1837


�

-series

� ��

� ��

� � �

Functional equation (for � primitive):

� ��

� � � ��

with � �� or � �� according as � � � � � �

.

– p.20/45

DIRICHLET, 1837


�

-series

� ��

� ��

� � �

Dirichlet’s Theorem: Every arithmetic progression

� , � �� , � � �� , ... without fixed divisor containsinfinitely many prime numbers.

– p.20/45

BASIC PROBLEMS ON PRIMES

Is there a simple formula for primes?

How to count the number of primes up to a givenbound?

How can we tell that a number is prime?

How can we find a prime factor of a compositenumber?

– p.21/45

COUNTING PRIMES

Euler and Gauss: The number of primesup to is about .

Legendre: .

Cebyshev (1848): If has a simpleapproximation, then

Gauss, letter to Hencke (1849).

– p.22/45

COUNTING PRIMES

Euler and Gauss: The number � ��

of primesup to � is about

�� .

Legendre: .



– p.22/45

COUNTING PRIMES



�� .

Legendre: � ��

� � �� .



– p.22/45

COUNTING PRIMES



�� .


� � �� .

Cebyshev (1848): If � �� has a simple

approximation, then

� ��

� ��

� � �

� ��

� � �

� � � � � � � � �


– p.22/45

COUNTING PRIMES



�� .


� � �� .

Cebyshev (1848): If � �� has a simple

approximation, then

� ��

� ��

� � �

� ��

� � �

� � � � � � � � �

Gauss, letter to Hencke (1849)� ��

� ��

�� .

– p.22/45

RIEMANN (1826-1866)

Riemannfounder of global analysis

– p.23/45

RIEMANN AND �

Riemann’s First Formula:

X

Y

The loop L

Implies the functional equation by deforming the loop.

Riemann’s Second Formula:

– p.24/45

RIEMANN AND �


� ��

� � � ��

� � � �

X

Y

The loop L



– p.24/45

RIEMANN AND �


� ��

� � � ��

� � � �

X

Y

The loop L



� � ��

� � � �

��

��

� � � ��

– p.24/45

RIEMANN AND � , II

Functional equation, new form:

The Logarithmic Derivative:

– p.25/45



� � ��

� � � � � � �

� ��

� � � � �

The Logarithmic Derivative:

– p.25/45



� � ��

� � � � � � �

� ��

� � � � �

The Logarithmic Derivative:� � � �

��

� � �

– p.25/45

RIEMANN AND � , III

Theorem: has

complex zeros with .

Riemann’s Assertion: Almost all complex zerosof have real part .

Riemann’s Conjecture: All complex zeros ofhave real part .

– p.26/45


Theorem:

� ��

has

��

� ��

��

��

complex zeros with

� � � �� .

Riemann’s Assertion: Almost all complex zerosof have real part .


– p.26/45


Theorem:

� ��

has

��

� ��

��

��

complex zeros with

� � � �� .

Riemann’s Assertion: Almost all complex zerosof

� ��

have real part�

� .


– p.26/45


Theorem:

� ��

has

��

� ��

��

��

complex zeros with

� � � �� .

Riemann’s Assertion: Almost all complex zerosof

� ��

have real part�

� .

Riemann’s Conjecture: All complex zeros of

� ��

have real part

�� .

– p.26/45

RIEMANN AND PRIMES

Riemann’s Theorem: If is not a prime power then

where runs over the complex zeros of and

Theorem:

where is the right-hand side of Riemann’s Theorem.

– p.27/45

RIEMANN AND PRIMES

Riemann’s Theorem: If � � �

is not a prime power then

��

��

� � � � � � �

�

� �

� � � � ��

where

�

runs over the complex zeros of� ��

and

� � �� ! � �" �� ! #$&% ' �

( � � �! $&% ' �) * �,+

+ �&- .0/ 1 �2 � � 3 4

Theorem:

where is the right-hand side of Riemann’s Theorem.

– p.27/45

RIEMANN AND PRIMES

Riemann’s Theorem: If � � �

is not a prime power then

��

��

� � � � � � �

�

� �

� � � � ��

where

�

runs over the complex zeros of� ��

and

� � �� ! � �" �� ! #$&% ' �

( � � �! $&% ' �) * �,+

+ �&- .0/ 1 �2 � � 3 4

Theorem:

� � � ��

��

� ��

� � � ��

where�

��

is the right-hand side of Riemann’s Theorem.

– p.27/45

A SIMPLER FORMULA

Cebyshev’s counting: It is better to count not just primes ,

but also prime powers , with weight:

if , and otherwise.

Theorem: Define . Then for not a prime

power we have

Theorem: .

– p.28/45

A SIMPLER FORMULA

Cebyshev’s counting: It is better to count not just primes � ,

but also prime powers � �

, with weight:

� � � � � � � � if � � � �

, and�

otherwise.

Theorem: Define . Then for not a prime

power we have

Theorem: .

– p.28/45

A SIMPLER FORMULA



, with weight:

� � � � � � � � if � � � �

, and�

otherwise.

Theorem: Define

� � � ��

� � �. Then for � � �

not a prime

power we have

� � � � � ��

� ��

��

� � � � � � � ( � � � � � � � � � 4

Theorem: .

– p.28/45

A SIMPLER FORMULA



, with weight:

� � � � � � � � if � � � �

, and�

otherwise.

Theorem: Define

� � � ��

� � �. Then for � � �

not a prime

power we have

� � � � � ��

� ��

��

� � � � � � � ( � � � � � � � � � 4

Theorem:� � � � � � � � 4 � 4 � 4 � ��

�� 4 4 4 � � � � .

– p.28/45

IS IT A SOLUTION?

Theorem: Define

� ��

��

. Then for � � not a prime

power we have

� ��

� ��

�

��

The zeros: with ,, , ...

graph of approximation: 50 zeros the overlap

– p.29/45

IS IT A SOLUTION?

The zeros:

�� with � � �� ,� � � � � � � � � � � � � � � � ,

�� , ...

graph of approximation: 50 zeros the overlap

– p.29/45

IS IT A SOLUTION?

The zeros:

�� with � � �� ,� � � � � � � � � � � � � � � � ,

�� , ...

graph of

� ��

approximation: 50 zeros the overlap

– p.29/45

THE CORRECTION TERMS

� ��

� � ��

.

The real part of a zero controls the growth

of the correction term . The imaginary part controlsthe period of oscillations.

, , ,

– p.30/45


� ��

� � ��

.

The real part

�

of a zero � � � �� controls the growth

of the correction term � � �� . The imaginary part controls

the period of oscillations.

, , ,

– p.30/45


� ��

� � ��

.

The real part

�

of a zero � � � �� controls the growth

of the correction term � � �� . The imaginary part controls

the period of oscillations.

� � � � � �� , �� , � � � � � � � � � � � � � � �� , � � ��

– p.30/45

THE PRIME NUMBER THEOREM

Theorem (Hadamard, de la Vallee-Poussin, 1896):.

Theorem (de la Vallee-Poussin, 1896):.

Theorem (von Koch, 1901): On the RiemannHypothesis

.

– p.31/45


Theorem (Hadamard, de la Vallee-Poussin, 1896):� �� .

Theorem (de la Vallee-Poussin, 1896):.


.

– p.31/45



Theorem (de la Vallee-Poussin, 1896):

� ��

��

.


.

– p.31/45



Theorem (de la Vallee-Poussin, 1896):

� ��

��

.


� ��

��

.

– p.31/45

THE PRIME NUMBER THEOREM, II

Approximation #1: .

Approximation #2: .

Examples: ,.

Deviation: ,,

.

– p.32/45


Approximation #1: � ��

�� .

Approximation #2: .

Examples: ,.

Deviation: ,,

.

– p.32/45



�� .


��

��

� �

� � �

.

Examples: ,.

Deviation: ,,

.

– p.32/45



�� .


��

��

� �

� � �

.

Examples: � ��

,� ��

.

Deviation: ,,

.

– p.32/45



�� .


��

��

� �

� � �

.

Examples: � ��

,� ��

.

Deviation: � � � � � ��

� � � � ��

� � � �

�� ,

� ��

��

� � � � ��

��

��

� � � ��

� � ��

� ��

� �� ,

� ��

� � � � � ��

� � � � � � � ��

� ��

� ��

��

� � � � � ��

��

� �

� � � � � � ��

�� .

– p.32/45

LARGE DEVIATIONS

Numerical evidence: for .

Theorem (Littlewood, 1914): The difference

changes sign infinitely often.

Skewes, 1955: There is a sign change before

.

Bays & Hudson, 2000: There is a sign change near.

– p.33/45

LARGE DEVIATIONS

Numerical evidence: � ��

�� for � � � �

.

Theorem (Littlewood, 1914): The difference



.


– p.33/45

LARGE DEVIATIONS


�� for � � � �

.

Theorem (Littlewood, 1914): The difference� ��

��



.


– p.33/45

LARGE DEVIATIONS


�� for � � � �

.


��



� � � � � ��

.


– p.33/45

LARGE DEVIATIONS


�� for � � � �

.


��



� � � � � ��

.

Bays & Hudson, 2000: There is a sign change near

��

� � � � � ��.

– p.33/45

FORMULAS FOR PRIMES?

Fermat’s claim: is always a prime.

Remark: True for , false for. (Euler,1732): divides .

Remark: verifies the Chinese test: divides(very easy to prove).

Gauss’s Theorem: A regular polygon with sidescan be constructed with ruler and compass if and onlyif is a power of times a product of distinct Fermatprimes.

– p.34/45


Fermat’s claim: �

� � � � � �

is always a prime.

Remark: True for , false for. (Euler,1732): divides .



– p.34/45



� � � � � �

is always a prime.

Remark: True for � � ��

��

��

��

�

, false for

� � ��

� � � � �

� �

. (Euler,1732): ��

divides � .



– p.34/45



� � � � � �

is always a prime.


��

��

��

�

, false for

� � ��

� � � � �

� �

. (Euler,1732): ��

divides � .

Remark: � verifies the Chinese test: � divides

� ��

��

(very easy to prove).


– p.34/45



� � � � � �

is always a prime.


��

��

��

�

, false for

� � ��

� � � � �

� �

. (Euler,1732): ��

divides � .

Remark: � verifies the Chinese test: � divides

� ��

��

(very easy to prove).

Gauss’s Theorem: A regular polygon with � sidescan be constructed with ruler and compass if and onlyif � is a power of

�

times a product of distinct Fermatprimes.

– p.34/45

FORMULAS FOR PRIMES, II

Problem: Find a "simple" formula for the-th prime.

Problem: Find a "simple" giving alwaysprimes.

Problem: Find a "simple" giving infinitelymany primes.

Problem: Find "simple" algorithms producingprimes (some, all, special types).

– p.35/45


Problem: Find a "simple" formula�

��

for the

� -th prime.

Problem: Find a "simple" giving alwaysprimes.



– p.35/45



��

for the

� -th prime.

Problem: Find a "simple"�

��

giving alwaysprimes.



– p.35/45



��

for the

� -th prime.

Problem: Find a "simple"�

��

giving alwaysprimes.

Problem: Find a "simple"

��

�

giving infinitelymany primes.


– p.35/45

CURIOSITIES?

Euler (1772): is prime for.

– p.36/45

CURIOSITIES?

� ��

��


– p.36/45

CURIOSITIES?

� ��

��

��

� � ��


– p.36/45

CURIOSITIES?

� ��

��

��

� � ��

Euler (1772): � � � �

is prime for� � ��

��

��.

– p.36/45

THE EQUATION � � � � � � � � �

Polynomials in one variable:

Arithmetic progressions (Dirichlet)

Conjecture: If is a polynomial without fixed divisor, then

infinitely often.

Conjecture: where

.

Polynomials in two variables:

Quadratic forms (Dirichlet)

Quadratic polynomials (Hooley, Linnik, Bredihin, ...)

(Friedlander and Iwaniec)

Cubic forms (Heath-Brown)

– p.37/45

THE EQUATION � � � � � � � � �



Conjecture: If

� ��

is a polynomial without fixed divisor, then

� �� infinitely often.

Conjecture: where

.






– p.37/45

THE EQUATION � � � � � � � � �



Conjecture: If

� ��



Conjecture: � �� where

� ��

.






– p.37/45

THE EQUATION � � � � � � � � �



Conjecture: If

� ��




� ��

.




��

��



– p.37/45

THE EQUATION � � � � � � � � �



Conjecture: If

� ��




� ��

.




��

��


Cubic forms (Heath-Brown)– p.37/45

PRIME RECOGNITION

The Wilson test: is prime if and only if it divides .

Gives a primality certificate. Not practical. There is no simple way

to compute the factorial.

The Fermat test (Fermat, Euler): If is prime then

divides for every integer coprime with .

Does not give a primality certificate, but is practical to compute.

Carmichael numbers: , , , ... are composite

but verify the condition.

Theorem (Alford, Granville, Pomerance): There are infinitely

many Carmichael numbers.

– p.38/45

PRIME RECOGNITION

The Wilson test: � is prime if and only if it divides� � �

� ��

.



The Fermat test (Fermat, Euler): If is prime then

divides for every integer coprime with .






– p.38/45

PRIME RECOGNITION


� ��

.



The Fermat test (Fermat, Euler): If � is prime then

� divides

��

��

for every integer�

coprime with � .






– p.38/45

PRIME RECOGNITION


� ��

.




� divides

��

��


coprime with � .


Carmichael numbers:� � �

,

� � � �

,

� ��

, ... are composite




– p.38/45

PRIME RECOGNITION


� ��

.




� divides

��

��


coprime with � .


Carmichael numbers:� � �

,

� � � �

,

� ��

, ... are composite



many Carmichael numbers.– p.38/45

PRIME RECOGNITION, II

Improved test:

If is an odd prime and , are coprime then divides

for an easily computable choice of sign .

The improvement: Suppose for example is

a product of two distinct odd primes and and divides

.

Then has four divisors , , , . One of them divides

exactly . In two cases out of four, this divisor

will be or and the improved primality test willdetect that is composite.

– p.39/45


Improved test:

If � is an odd prime and

�

, � are coprime then � divides

� ��

for an easily computable choice of sign

.

The improvement: Suppose for example is

a product of two distinct odd primes and and divides

.

Then has four divisors , , , . One of them divides

exactly . In two cases out of four, this divisor

will be or and the improved primality test willdetect that is composite.

– p.39/45


Improved test:

If � is an odd prime and

�

, � are coprime then � divides

� ��

for an easily computable choice of sign

.

The improvement: Suppose for example � � �� is

a product of two distinct odd primes � and � and divides

��

.

Then � has four divisors�

, � , � , � . One of them divides

exactly

� � � � � �� . In two cases out of four, this divisor

will be � or � and the improved primality test willdetect that � is composite.

– p.39/45

PRIME RECOGNITION, III

The Miller-Bach test:

Assume that a generalized Riemann hypothesis holds.

Do the improved test for every .

If passes the tests then is prime.

Gives a fast primality certificate but only on the assumption of a unproven

hypothesis (there is a $1000000 bounty for its solution).

The Rabin probabilistic test: Do the improved test choosing

at random times. Then the probability that the test gives you

a false positive is less than the probability of hardware or software

failure in your computer in running the program.

Very good in practice, but not giving a primality certificate.

– p.40/45




Do the improved test for every

� � �

� � ��

.

If � passes the tests then � is prime.




at random times. Then the probability that the test gives you




– p.40/45




Do the improved test for every

� � �

� � ��

.

If � passes the tests then � is prime.




�

at random

� � �

times. Then the probability that the test gives you




– p.40/45

HOW FAST IS YOUR PROGRAM?

Fast – Polynomial complexity:

Running time: comparable to a power of the size of the input

(e.g. bit length).

Example: Sum: number of steps is about bit length.

Example: Multiplication: number of steps is about the square of bit length.

Example: Multiplication: using FFT, multiplication is steps

( ). Very useful in practice (CAT scan, oil exploration).

Slow – Exponential complexity:

Running time is exponential in the number of digits.

Example: Analyzing moves in a chess game is exponential in

the depth level of analysis.

– p.41/45

HOW FAST IS YOUR PROGRAM?

Fast – Polynomial complexity:

Running time: comparable to a power of the size of the input

(e.g. bit length).

Example: Sum: number of steps is about bit length.

Example: Multiplication: number of steps is about the square of bit length.

Example: Multiplication: using FFT, multiplication is

� ��

steps

( � ��

). Very useful in practice (CAT scan, oil exploration).

Slow – Exponential complexity:

Running time is exponential in the number of digits.

Example: Analyzing moves in a chess game is exponential in

the depth level of analysis.

– p.41/45

PRIME RECOGNITION, IV

The new test:

Theorem (Agrawal - Kayal - Saxena, 2002) Let be a prime not

dividing and such that has a large prime factor

and moreover does not divide .

Then is prime if and only if

is divisible by for every , coprime with .

Theorem (Goldfeld, Fouvry,... (1969,1985)): We can find and

with the right conditions and .

Hence the AKS algorithm runs in polynomial time.

– p.42/45


The new test:

Theorem (Agrawal - Kayal - Saxena, 2002) Let � be a prime not

dividing � and such that � � � has a large prime factor

� � � � ��

and moreover � does not divide � �� .

Then � is prime if and only if

��

� � ��

is divisible by � for every� � � � , coprime with � .

Theorem (Goldfeld, Fouvry,... (1969,1985)): We can find and

with the right conditions and .


– p.42/45


The new test:

Theorem (Agrawal - Kayal - Saxena, 2002) Let � be a prime not

dividing � and such that � � � has a large prime factor

� � � � ��

and moreover � does not divide � �� .

Then � is prime if and only if

��

� � ��

is divisible by � for every� � � � , coprime with � .

Theorem (Goldfeld, Fouvry,... (1969,1985)): We can find � and �

with the right conditions and � � � � � � � �

.


– p.42/45

APPLICATIONS

Public Key Encryption: A method in which the knowledge

of the encoding key does not help in decoding.The public key can also be changed for every message.

A bit like the trap door: Knowing how you fall in the

dungeon does not help you to exit out of it.

The Protocol: If (uyer) buys an item on the Internet from

(endor), he needs to send a secure (essage) with

personal data (item, delivery address, credit card number).

receives from an encryption rule or (ey). The

encrypted message is sent back to , who is the

only one to know the inverse key and compute

back .

– p.43/45

APPLICATIONS

Public Key Encryption: A method in which the knowledge

of the encoding key does not help in decoding.The public key can also be changed for every message.

A bit like the trap door: Knowing how you fall in the

dungeon does not help you to exit out of it.

The Protocol: If

�

(uyer) buys an item on the Internet from

�

(endor), he needs to send a secure (essage) with

personal data (item, delivery address, credit card number).

�

receives from�

an encryption rule or

�

(ey). The

encrypted message

� � �

is sent back to

�

, who is the

only one to know the inverse key

� � �

and compute

back� � � � � � � � � .

– p.43/45

FACTORIZATION

Importance of factorization: Many methods of public key

encryption depend on knowing the factorization of

a number product of two primes.

Fast factorization of very large numbers ( digits)would create serious vulnerability of the Internet network.

Complexity today: Less than exponential, but far from

polynomial (best ).

A famous lecture (F.N. Cole, 1903):

– p.44/45

FACTORIZATION



a number

�� product of two primes.

Fast factorization of very large numbers (��

digits)would create serious vulnerability of the Internet network.


polynomial (best ).


– p.44/45

FACTORIZATION



a number





polynomial (best � � � ��

).


– p.44/45

FACTORIZATION



a number






).


� � ��

– p.44/45

FACTORIZATION



a number






).


� � ��

� � � ��

– p.44/45

THE PRESS ON AKS

New York Times, Aug. 8: “New Method Said to Solve Key Problem in Math”

Website of the Tagesschau, Aug 12: “At last prime numbers can be exactly

calculated!”, “The joy at German schools is boundless: finally one can

calculate prime numbers without tears!”

Removed after several protests were received.

The Wall Street Journal, Nov. 4: “One beautiful mind from India is putting

the Internet on alert”

The Neue Zürcher Zeitung, Aug. 30: The article suggested (wrongly) that

until now no prime certification could be given within a reasonable time for

primes used in cryptography. The result was not so well considered by

the media because it could not handle the largest known prime number.

Fact (Cameron, 2001): is a prime with

digits.

– p.45/45

THE PRESS ON AKS

New York Times, Aug. 8: “New Method Said to Solve Key Problem in Math”

Website of the Tagesschau, Aug 12: “At last prime numbers can be exactly

calculated!”, “The joy at German schools is boundless: finally one can

calculate prime numbers without tears!”

Removed after several protests were received.

The Wall Street Journal, Nov. 4: “One beautiful mind from India is putting

the Internet on alert”

The Neue Zürcher Zeitung, Aug. 30: The article suggested (wrongly) that

until now no prime certification could be given within a reasonable time for

primes used in cryptography. The result was not so well considered by

the media because it could not handle the largest known prime number.

Fact (Cameron, 2001):

� � � � ��

is a prime with

��

� � ��

� �

digits. – p.45/45

the 2003 pauli lectures - eth zürich - homepageeuler (1707-1783) euler’s father wanted his son to...

Documents