STUDENT MATHEMAT ICAL L IBRARYVolume 70

Primality Testing for Beginners

Lasse Rempe-Gillen

Rebecca Waldecker

Primality Testing for Beginners

http://dx.doi.org/10.1090/stml/070

Primality Testing for Beginners

Lasse Rempe-Gillen Rebecca Waldecker

Student mathemat ical l ibraryVolume 70

American Mathematical SocietyProvidence, Rhode Island

Editorial Board

Satyan L. DevadossGerald B. Folland (Chair)

John StillwellSerge Tabachnikov

The cover illustration is a variant of the Sieve of Eratosthenes (Sec-tion 1.5), showing the integers from 1 to 2704 colored by the number oftheir prime factors, including repeats. The illustration was created us-ing MATLAB. The back cover shows a phase plot of the Riemann zetafunction (see Appendix A), which appears courtesy of Elias Wegert(www.visual.wegert.com).

2010 Mathematics Subject Classification. Primary 11-01, 11-02, 11Axx,11Y11, 11Y16.

For additional information and updates on this book, visitwww.ams.org/bookpages/stml-70

Library of Congress Cataloging-in-Publication Data

Rempe-Gillen, Lasse, 1978– author.[Primzahltests fur Einsteiger. English]Primality testing for beginners / Lasse Rempe-Gillen, Rebecca Waldecker.

pages cm. — (Student mathematical library ; volume 70)Translation of: Primzahltests fur Einsteiger : Zahlentheorie - Algorithmik -

Kryptographie.Includes bibliographical references and index.ISBN 978-0-8218-9883-3 (alk. paper)1. Number theory. I. Waldecker, Rebecca, 1979– author. II. Title.

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10 9 8 7 6 5 4 3 2 1 18 17 16 15 14

For our spouses,

Emma and Lars

Contents

Preface xi

Introduction 1

Part 1. Foundations

Chapter 1. Natural numbers and primes 13

§1.1. The natural numbers 13

§1.2. Divisibility and primes 26

§1.3. Prime factor decomposition 31

§1.4. The Euclidean algorithm 35

§1.5. The Sieve of Eratosthenes 39

§1.6. There are infinitely many primes 41

Further reading 42

Chapter 2. Algorithms and complexity 43

§2.1. Algorithms 43

§2.2. Decidable and undecidable problems 52

§2.3. Complexity of algorithms and the class P 57

§2.4. The class NP 68

vii

viii Contents

§2.5. Randomized algorithms 73

Further reading 81

Chapter 3. Foundations of number theory 83

§3.1. Modular arithmetic 84

§3.2. Fermat’s Little Theorem 94

§3.3. A first primality test 104

§3.4. Polynomials 107

§3.5. Polynomials and modular arithmetic 120

Further reading 127

Chapter 4. Prime numbers and cryptography 129

§4.1. Cryptography 129

§4.2. RSA 132

§4.3. Distribution of primes 136

§4.4. Proof of the weak prime number theorem 139

§4.5. Randomized primality tests 143

Further reading 150

Part 2. The AKS Algorithm

Chapter 5. The starting point: Fermat for polynomials 153

§5.1. A generalization of Fermat’s Theorem 153

§5.2. The idea of the AKS algorithm 159

§5.3. The Agrawal-Biswas test 163

Chapter 6. The theorem of Agrawal, Kayal, and Saxena 169

§6.1. Statement of the theorem 170

§6.2. The idea of the proof 171

§6.3. The number of polynomials in P 173

§6.4. Cyclotomic polynomials 178

Contents ix

Chapter 7. The algorithm 183

§7.1. How quickly does the order of n modulo r grow? 183

§7.2. The algorithm of Agrawal, Kayal, and Saxena 186

§7.3. Further comments 189

Further reading 192

Appendix A. Open questions 193

Further reading 205

Appendix B. Solutions and comments to important exercises 207

Bibliography 233

List of symbols 237

Index 239

Preface

“How on earth can you do research in mathematics?”, we are of-

ten asked. The idea of ongoing research in other sciences is deeply

embedded in the public consciousness, not only in those fields that

may lead to self-driving cars or new life-saving drugs, but also in

the most theoretical areas such as particle physics, where scientists

try to puzzle out the likely nature of matter on the smallest possible

scales. In contrast, even among members of the public who profess

their enjoyment of mathematics there is little awareness that many

mathematical questions remain open and that these are the subject

of intensive investigation.

One of the difficulties in challenging this perception lies in the

highly specialized nature of modern mathematics itself. Even profes-

sional mathematicians are usually unable to fully appreciate research

advances outside their own branches of mathematics. Of course there

are exceptions to this rule, such as Fermat’s Last Theorem, which

was proved by Andrew Wiles in the 1990s and could hardly be sim-

pler to state: “If n > 2, then there are no non-zero integers a, b, and

c such that an + bn = cn.” Yet in this case the proof is particularly

long and difficult, and only a small number of experts worldwide are

able to fully comprehend it.

xi

xii Preface

In the summer of 2002, the computer scientist Manindra Agrawal

and his students Neeraj Kayal and Nitin Saxena achieved a remark-

able feat: they discovered an efficient and deterministic test for the

primality of a natural number. (To learn about the meaning of these

notions, keep reading!) We were fascinated by this result not only be-

cause it answers a long-standing open question, but because the math-

ematics behind it is beautiful and, compared to other modern research

advances, extremely accessible. We decided to use this opportunity

to get young people in touch with actual mathematical research by

offering a course on the subject at a German summer academy for

secondary-school students in 2005. Inspired by the participants and

their enthusiasm, we started to transform the course material into

a book manuscript, the German version of which appeared in 2009.

The book you hold in your hands is an English-language edition of

this text. During the translation process, we have also corrected a

few errors, changed some of the presentation for pedagogical reasons,

and updated the content with new results, where appropriate.

We thank the students from our original summer course for their

motivation and dedication and all those who have since helped us

for their comments and suggestions regarding both the German and

English editions. We also thank you – our reader – for your interest in

this little volume, and we hope that you will enjoy reading the book

as much as we did writing it.

Lasse Rempe-Gillen and Rebecca Waldecker

August 2013

Bibliography

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[AKS] Agrawal, M., Kayal, N., and Saxena, K.: PRIMES is in P. Annalsof Math. 160 (2004), no. 2, 781–793.

[AO] Agargun, A. Goksel and Ozkan, E. Mehmet: A historical survey ofthe fundamental theorem of arithmetic. Historia Math. 28 (2001),no. 3, 207–214.

[Be] Beutelspacher, A.: Cryptology. Mathematical Association ofAmerica, 1994.

[Bo] Bornemann, F.: Primes is in P: A breakthrough for “Everyman”.Notices of the AMS 50 (2003), no. 5, 545–552.

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[Ch] Chen, J. R.: On the representation of a larger even integer as thesum of a prime and the product of at most two primes. Sci. Sinica16 (1973), 157–176.

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[CLR] Cormen, T. H., Leiserson, C. E., and Rivest, R. L.: Introductionto Algorithms. Third Edition, MIT Press, 2009.

[CM] Coron, J.-S. and May, A.: Deterministic polynomial-time equiva-lence of computing the RSA secret key and factoring. J. Cryptology20 (2007), 39–50.

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[De] Derbyshire, J.: Prime Obsession: Bernhard Riemann and theGreatest Unsolved Problem in Mathematics. Penguin, 2004.

[Dst] Diestel, R.: Graph Theory. Fourth edition, Springer, 2010.

[Dtz] Dietzfelbinger, M.: Primality Testing in Polynomial Time: FromRandomized Algorithms to “PRIMES Is in P”. Springer, 2004.

[Do] Doxiadis, A.K.: Uncle Petros and Goldbach’s Conjecture: A Novelof Mathematical Obsession. Bloomsbury, 2001.

[EFT] Ebbinghaus, H.-D., Flum, J., and Thomas, W.: MathematicalLogic. Second edition, Springer, 1994.

[Eb] Ebbinghaus et al.: Numbers. Third printing, Springer, 1996.

[Ec] Eccles, P.: An Introduction to Mathematical Reasoning: Numbers,Sets and Functions. Cambridge University Press, 1997.

[Frl] Fraleigh, J. B.: A First Course in Abstract Algebra. Seventh edi-tion, Pearson, 2003.

[Frz] Franzen, T.: Godel’s Theorem: An Incomplete Guide to Its Useand Abuse. Peters, 2005.

[Fu] Furer, M.: Faster Integer Multiplication. Proceedings of the 39thAnnual ACM Symposium on Theory of Computing (2007), 57–66.

[G] Granville, A.: It is easy to determine whether a given integer isprime. Bull. Amer. Math. Soc. 42 (2005), no. 1, 3–38.

[GT] Green, B. and Tao, T.: The primes contain arbitrarily long arith-metic progressions.Annals of Math. (2) 167 (2008), no. 2, 481–547.

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[Har] Hardy, G. H.: A Mathematician’s Apology. Cambridge UniversityPress, 1940.

[HW] Hardy, G. H. and Wright, E. M.: An Introduction to the Theoryof Numbers. Oxford University Press, 2008.

[He] Helfgott, H. A.: Major arcs for Goldbach’s problem, preprint,arXiv:1305.2897, 2013.

[Ho] Hofstadter, D.: Godel, Escher, Bach: An Eternal Golden Braid.Penguin Books Ltd., 1980.

[HMU] Hopcroft, J. E., Motwani, R., and Ullman, J. D.: Intro-duction to Automata Theory, Languages, and Computation.Pearson/Addison-Wesley, 2007.

[HP] Humphreys, J. F. and Prest, M. Y.: Numbers, Groups and Codes.Second edition, Cambridge University Press, 2004.

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[J] Jameson, G. J. O.: The Prime Number Theorem. Cambridge Uni-versity Press, 2008.

[Ke] Kelly, T.: The myth of the Skytale. CRYPTOLOGICA, Vol. XXII,no. 3 (1998).

[KS] Kurzweil, H. and Stellmacher, B.: The Theory of Finite Groups.Springer, 2004.

[LaPe] Laubenbacher, R. and Pengelley, D.: “Voici ce que j’ai trouve”:Sophie Germain’s grand plan to solve Fermat’s Last Theorem. His-toria Mathematica 37, Issue 4, (2010), 641–692.

[LP] Lenstra, H. W. Jr. and Pomerance, C.: Primality testing withGaussian periods. Preprint, 2005, revised April 2011.http://www.math.dartmouth.edu/∼carlp/aks041411.pdf.

[LPa] Lewis, H. R. and Papadimitriou, C. H.: Elements of the Theory ofComputation. Prentice Hall International, 1998.

[LiN] Lidl, R. and Niederreiter, H.: Finite Fields. Addison-Wesley, 1983.

[Lo] Lorenz, F.: Algebra. Volume I: Fields and Galois Theory. Springer,2006.

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[N] Nair, M.: On Chebyshev-type inequalities for primes. Amer. Math.Monthly 89, no. 2 (1982), 126–129.

[NZM] Niven, I., Zuckerman, H. S., and Montgomery, H. L.: An Intro-duction to the Theory of Numbers. John Wiley & Sons, 1991.

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[Za] Zagier, D.: Newman’s short proof of the prime number theorem.American Math. Monthly 104 (1997), 705-708.

[Zh] Zhang, Y.: Bounded gaps between primes. Annals of Math., toappear.

List of symbols

∅ (the empty set), page 8

C (the complex numbers), page 203

N (the natural numbers, not including zero), page 7

N0 (the natural numbers, including zero), page 7

Q (the rational numbers), page 7

R (the real numbers), page 7

Z (the integers), page 7

∞ (infinity), page 8

e (Euler’s constant, e = 2,71828 . . . ), page 9

π (the area of a circle of radius 1, π = 3.14159 . . . ), page 48

#M (number of elements of M), page 8

N ⊆ M (N is a subset of M), page 8

N � M (N is a proper subset of M), page 8

x ∈ M (x is an element of M), page 8

y /∈ M (y is not an element of M), page 8

NP (efficiently verifiable problems), page 69

P (efficiently computable problems), page 60

RP (problems with efficient Monte Carlo solutions), page 75

ZPP (problems with efficient Las Vegas solutions), page 78

f : N → M (f is a function from N to M), page 9

f(n) = O(g(n)) (f grows asymptotically at most as quickly as g), page 59

:= (is defined as), page 8

a ≤ b (a is less than or equal to b), page 7

a < b (a is strictly less than b), page 7

lnx (the natural logarithm of x), page 9

log x (the logarithm of x to base 2), page 9∏n

i=1 xi (the product x1 · x2 · · · xn), page 8

237

238 List of symbols

∑ni=1 xi (the sum x1 + x2 + · · · + xn), page 8

�x� (the largest integer n ≤ x), page 9

�x (the smallest integer n ≥ x), page 9

|x| (absolute value of x), page 9

n! (n factorial), page 9

k |n (k divides n), page 26

gcd(a, b) (the greatest common divisor of a and b), page 27

lcm(a, b) (the least common multiple of a and b), page 27

cp(n) (the set of integers from 1 to n−1 that are coprime to n), page 98

ϕ(n) (the number of integers from 1 to n − 1 that are coprime to n),page 98

π(n) (the number of primes ≤ n), page 137

M mod n (the remainder of M after division by n), page 86

a ≡ b (mod n) (a is congruent to b modulo n), page 84

ordn(a) (the order of a modulo n), page 94

degP (the degree of the polynomial P ), page 108

degn(P ) (the degree of P modulo n), page 121

P ≡ Q (mod H) (P and Q are congruent modulo H), page 113

P ≡ Q (mod n) (P and Q are congruent modulo n), page 120

P ≡ Q (mod n,H) (P and Q are congruent modulo n and H), page 122

P (the set of all integer polynomials P satisfying (6.1.2)), page 171

r(n, k) (the smallest prime r with r |n or ordr(n) > k), page 184

Index

3n+ 1 problem, 51

Aaronson, Scott, 73

absolute value, 9

Ackermann, Wilhelm

Ackermann function, 68

Adleman, Leonard, 1, 132, 135,192

age of the universe, 40

Agrawal, Manindra, 163

AKS algorithm, 4, 186

al-Khwarizmi, Abu AbdallahMuhammad ibn Musa, 50

Alberti, Leon Battista, 131

algorithm, 43, 45

ADDITION, 46, 61

AKS, 4, 186

COLLATZ, 51

CRIME-BESTSELLER, 44

deterministic, 73

efficient, 60

Euclidean, 29, 35, 36, 43, 50, 66

inefficient, 40

Karatsuba, 67

Las Vegas, 77

MILLER-RABIN, 145

Monte Carlo, 75

N*(PI+E), 48

of Agrawal, Kayal, and Saxena,4, 186

PANCAKE, 44

QUICKSORT, 76

randomized, 73

Schonhage-Strassen, 68

analysis

mathematical, 143

Annals of Mathematics, 4, 25, 199

Arenstorf, Richard F., 204

arithmetic

modular, 84

arithmetic progression, 199

asymptotic growth, 59

asymptotic running time, 60

average running time, 78

axiom, 14

Peano axioms, 25

Bachmann, Paul, 66

base, 104

basis of the induction, 17

best-selling crime novel, 44

Bezout, Etienne

Bezout’s Lemma, 37, 39, 87

binary number system, 46, 51

binomial coefficients, 19

explicit formula, 23

recursive formula, 19

binomial theorem, 19

Biswas, Somenath, 163

bounded from above/below, 22

239

240 Index

Caesar, Julius, 66

Caesar cipher, 130

calculus

integral/differential, 143

Carmichael, Robert D.Carmichael number, 106, 107,

146Chebyshev, Pafnuty Lvovich, 139

Chen, Jingrun, 195, 198

Chinese Remainder Theorem, 89

Church, Alonzo, 50, 54

Church-Turing thesis, 50

cipher

digraphic substitution, 131

mono-alphabetic, 130

playfair, 131

poly-alphabetic, 131

RSA, 132

ciphertext, 130

Clement’s Theorem, 199

Clement, Paul A.

Clement’s Theorem, 199

coefficient, 108

leading, 108

Collatz Conjecture, 51Collatz, Lothar, 51

colorability, 70

combinatorics, 19

common divisor, 27

greatest, 27

common factor, 27

highest, 27

common multiple, 27

least, 27

complete set of residues, 92

complex numbers, 181, 193, 203

complex analysis, 139

complexity theory, 58

composite number, 26

COMPOSITES, 53

conclusion, 6

congruence, 84

modulo a polynomial, 113

of polynomials modulo n, 120congruence class, 93

congruent, 84

constant polynomial, 108

contradiction

proof by, 15

converse, 7

coprime, 27

corollary, 7

coset, 100

counterexample

smallest, 15

cryptography, 129

public-key, 132

CSR, 92

cyclotomic polynomial, 178

de la Vallee Poussin,Charles-Jean, 139

de Vigenere, Blaise, 131

decidable problem, 52

decimal number system, 51

decision problem, 52

decomposition

into irreducible factors, 125

into prime factors, 31

defined as, 8

definition, 6

recursive, 18

degree of a polynomial, 108

modulo n, 121

descent

infinite, 15

determinism, 45

deterministic algorithm, 73

diagonalization, 57

differential calculus, 143

Diffie, Whitfield, 135

digraphic substitution ciphers, 131

Diophantine equation, 92

direct proof, 18

Dirichlet, Gustav Lejeune

Dirichlet’s Prime NumberTheorem, 200

Disquisitiones Arithmetica, 34

divide and conquer, 62

division

long, 29

polynomial long division, 110

with remainder, 28

division algorithm, 28

division theorem, 28

division with remainder, 28

Index 241

for polynomials, 111

divisor, 26

common, 27

greatest common, 27

non-trivial, 26

of a polynomial, 110, 112

of a polynomial modulo n, 122

trivial, 26

zero, 119

zero divisor, 91, 93

dual problem, 53

efficiency, 60

efficiently verifiable problem, 69

Eisenstein, Ferdinand, 203

irreducibility criterion, 126

elements of a set, 8

empty set, 8

encryption

RSA, 2

Entscheidungsproblem, 50, 54

Eratosthenes of Cyrene, 40

Sieve of, 39, 43, 61

Euclid, 34, 38, 41, 200

Euclid number, 200

Euclidean algorithm, 29, 35, 36, 43,50, 66

Euler, Leonhard, 194, 195

Euler’s constant, 9

Fermat-Euler Theorem, 99

phi function, 104

totient function, 98, 104

even integers, 7

even number, 22, 84

factorial, 9

Factoring Challenge, 2

Fermat number, 203

Fermat prime, 203

Fermat test, 105

Fermat, Pierre de, 2, 25

Fermat’s Last Theorem, xi, 25,189, 191

Fermat’s Little Theorem, 25, 96

Fermat-Euler Theorem, 99

Last Theorem of, 197

theorem of Fermat-Miller, 145

Fibonacci numbers, 19, 66

field, 93finite arithmetic progression, 199

Fouvry, Etienne, 190, 191

fraction, 7frequency analysis, 130function, 9

one-way, 132

Fundamental Theorem ofArithmetic, 31

Furer, Martin, 68

Gauss, Carl Friedrich, 104, 139Disquisitiones Arithmetica, 34

Gauss summation, 22gcd, 27Generalized Riemann Hypothesis,

149, 192, 195

Germain, Sophie, 189, 191, 202Godel, Kurt, 57

incompleteness theorem, 57Goldbach, Christian, 195

Goldbach Conjecture, 195Weak Goldbach Conjecture, 196

Goldberg’s Conjecture, 197graph, 56

Great Internet Mersenne PrimeSearch, 201

greatest common divisor, 27Green, Ben

Green-Tao Theorem, 199

group, 104

Hadamard, Jacques, 139halting problem, 54Hardy, Godfrey Harold, 1, 190,

205

Heath-Brown, David Rodney, 192Helfgott, Harald, 197Hellman, Martin, 135highest common factor, 27

Hilbert’s decision problem, 54Hilbert, David, 49, 194

Entscheidungsproblem, 50, 54Tenth Problem, 56

hypothesis, 6

inductionbasis of, 17mathematical, 16

242 Index

variants, 18, 25induction hypothesis, 17induction step, 17

inefficient algorithm, 40infinite descent, 15input, 52instance, 52

positive/negative, 53integer, 7

even, 7odd, 7

integral, 143integral calculus, 143integral domain, 119integral logarithm, 139

intractable problem, 60inverse modulo n, 87irreducible polynomial, 114

Eisenstein criterion, 126

modulo p, 123

Jacobi symbol, 149

Karatsuba, Anatoly AlexeevitchKaratsuba algorithm, 67

Kasiski, Friedrich W., 131

keyprivate, 132public, 132

Lagrange, Joseph-LouisLagrange’s Theorem, 101, 104

Landau, EdmundLandau symbols, 66

Las Vegas algorithm, 77Las Vegas method, 73lcm, 27leading coefficient, 108

least common multiple, 27Legendre symbol, 149Legendre, Adrien-Marie, 139lemma, 7

Bezout’s Lemma, 39, 87Lenstra, Hendrik W., 190Library of Alexandria, 40

linear factor, 112, 117, 124Littlewood, John E., 190, 205logarithm

base 2, 9

integral, 139

natural, 9

Logarithmic Integral Function, 193

long division, 29

polynomial, 110

Lucas, Edouard

Lucas test, 201

Markov, Andrei Andreyevich

Markov inequality, 78, 81

mathematical analysis, 143

mathematical induction, 16

variants, 18, 25

Mersenne prime

Great Internet Mersenne PrimeSearch, 201

Mersenne, Marin

Mersenne number, 200

Mersenne prime, 201

Millennium Prize Problems, 72, 194

Miller, Gary Lee, 3, 149

theorem of Fermat-Miller, 145

Miller-Rabin primality test, 145

modular arithmetic, 84

modulo a polynomial, 113

monic, 108

mono-alphabetic cipher, 130

Monte Carlo algorithm, 75

Monte Carlo method, 73

multiple, 17, 26

common, 27

least common, 27

multiplicative inverse, 87

National Lottery, 19

natural logarithm, 9

natural numbers, 7, 13

negative instance, 53

non-trivial divisor, 26

of a polynomial, 112

NP, 68, 69

NP-complete problem, 72

number

Carmichael, 106, 107

complex, 181, 193, 203

composite, 26

Euclid, 200

even/odd, 22, 84

Index 243

integer, 7

natural, 7, 13

perfect, 200

prime, 1, 26

rational, 7

real, 7

number of atoms in the universe,40, 68, 197

O-notation, 59, 66

odd integers, 7

odd number, 22, 84

one-way function, 132

order modulo n, 94

output, 52

P, 60

pairwise, 23

pancake, 44

particle physics, xi

Pascal’s triangle, 19, 96

Pausanias, 130

Peano axioms, 25

Peano, Giuseppe, 25

Peano axioms, 25

perfect number, 200

perfect power, 9

plaintext, 130

playfair cipher, 131

poly-alphabetic cipher, 131

Polybius square, 130

polynomial, 108

constant, 108

cyclotomic, 178

in X/Y /Z, 109

in several variables, 56, 73

integer/rational, 108

irreducible, 114

irreducible (modulo p), 123

monic, 108

over a field, 119

reducible, 118

zero, 108

polynomial running time, 60

polynomial zero, 116, 123

Pomerance, Carl, 190

positive instance, 53

Post, Emil

Correspondence Problem, 56

power

perfect, 9

Pratt, Vaughan, 72

prime, 1, 26

Fermat, 203

Mersenne, 201

root of unity modulo a prime,144

Sophie Germain prime, 190

twin, 42

prime factor, 30

prime factor decomposition, 31

prime gap, 41

prime number theorem, 137

PRIMES, 53

primitive root, 103

primitive root of unity, 181

private key, 132

problem, 52

3n+ 1, 51

class NP, 68, 69

class P, 60

class RP, 75class ZPP, 78

COMPOSITES, 53

decision, 52

dual, 53

efficiently computable, 60

efficiently verifiable, 69

halting problem, 54

Hilbert’s Entscheidungsproblem,50, 54

Hilbert’s Tenth Problem, 56

intractable, 60

NP-complete, 72

Post Correspondence Problem,56

PRIMES, 53

search, 52

SUM, 52

(un)decidable, 52

product notation, 8

proof, 6

by contradiction, 15

direct, 18

pseudo-prime, 105

strong, 146

244 Index

public key, 132public-key cryptography, 132

Quicksort, 76

Rabin, Michael Oser, 3, 149Ramare, Olivier, 196

randomized algorithm, 73rational numbers, 7real numbers, 7recursive definition, 18

reduced set of residues, 92reducible polynomial, 118Riemann ζ-function, 193Riemann Hypothesis, 139, 193

Generalized, 149, 192, 195, 197Riemann, Bernhard, 193Rivest, Ronald, 1, 132, 135root of unity, 181

modulo a prime, 144primitive, 181

RP, 75

RSA, 2, 132RSA number, 2RSR, 92running time

asymptotic, 60average, 78exponential, 60function, 58

polynomial, 60

Schonhage, Arnold, 68Schonhage-Strassen algorithm, 68

search problem, 52set, 8

uncountable, 57Shamir, Adi, 1, 132, 135

Sieve of Eratosthenes, 39, 43, 61Skytale, 130smallest counterexample, 15Solovay, Robert Martin, 149

Sophie Germain prime, 190, 202Strassen, Volker, 68, 149string, 52, 58

strong pseudo-prime, 146subset, 8successor, 24sufficiently large, 59

sum notation, 8Sylvester, James Joseph, 104

Tao, Terence, 199Green-Tao Theorem, 199

Tenenbaum, Gerald, 204theorem, 7theorem of Agrawal, Kayal, and

Saxena, 169theorem of Fermat-Miller, 145totient function, 98triples of prime numbers, 198trivial divisor, 26Turing, Alan, 50, 54, 131

twin primes, 42, 197

uncountable set, 57undecidable problem, 52unique modulo n, 88universe

age, 40number of atoms, 40, 68, 197

Vigenere square, 131Vinogradov, Ivan Matveyevich

Vinogradov’s Theorem, 196von Koch, Helge, 139, 194

Weak Goldbach Conjecture, 196well-ordering principle, 14What’s my line?, 62Wheatstone, Charles, 131Wiles, Andrew, xi, 25, 192Wilson’s Theorem, 149, 204Wilson, John, 149, 204witness, 69

zeropolynomial, 116, 123

zero divisor, 91, 93, 115, 119zero of a polynomial, 109zero polynomial, 108

Zhang, Yitang, 198ZPP, 78

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How can you tell whether a number is prime? What if the number has hundreds or thousands of digits? This question may seem abstract or irrelevant, but in fact, primality tests are performed every time we make a secure online transaction. In 2002, Agrawal, Kayal, and Saxena answered a long-standing open question in this context by presenting a deterministic test (the AKS algorithm) with polynomial running time that checks whether a number is prime or not. What is more, their methods are essentially elementary, providing us with a unique opportunity to give a complete explanation of a current mathematical breakthrough to a wide audience.

Rempe-Gillen and Waldecker introduce the aspects of number theory, algorithm theory, and cryptography that are relevant for the AKS algo-rithm and explain in detail why and how this test works. This book is specifi cally designed to make the reader familiar with the background that is necessary to appreciate the AKS algorithm and begins at a level that is suitable for secondary school students, teachers, and interested amateurs. Throughout the book, the reader becomes involved in the topic by means of numerous exercises.