Table Of ContentSelected Chapters of Number Theory: Special Numbers
Series Editor: Elena Deza (Moscow State Pedagogical University, Russia)
Published
Vol. 1 Mersenne Numbers and Fermat Numbers
by Elena Deza
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Selected Chapters of Number Theory: Special Numbers — Vol. 1
MERSENNE NUMBERS AND FERMAT NUMBERS
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Mathematics is the queen of the Sciences and
Number Theory is the queen of Mathematics.
— Gauss
God invented the integers; all else is the work of man.
— Kronecker
I know numbers are beautiful.
If they aren’t beautiful, nothing is.
— Erd˝os
v
B1948 Governing Asia
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Contents
Notations ix
Preface xv
1. Preliminaries 1
1.1 Divisibility of integers . . . . . . . . . . . . . . . . . 1
1.2 Modular arithmetics . . . . . . . . . . . . . . . . . . 8
1.3 Arithmetic functions . . . . . . . . . . . . . . . . . . 11
1.4 Solution of congruences . . . . . . . . . . . . . . . . 16
1.5 Quadratic residues, Legendre symbol and
Jacobi symbol . . . . . . . . . . . . . . . . . . . . . . 26
1.6 Multiplicative orders, primitive roots
and indexes . . . . . . . . . . . . . . . . . . . . . . . 29
1.7 Continued fractions . . . . . . . . . . . . . . . . . . . 33
2. Prime numbers 39
2.1 History of the question . . . . . . . . . . . . . . . . . 39
2.2 Elementary properties of prime numbers . . . . . . . 43
2.3 How to recognize whether a natural number
is a prime?. . . . . . . . . . . . . . . . . . . . . . . . 46
2.4 Formulas of primes . . . . . . . . . . . . . . . . . . . 62
2.5 Prime numbers in the family of special numbers . . . 74
2.6 Open problems . . . . . . . . . . . . . . . . . . . . . 87
vii
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viii Mersenne Numbers and Fermat Numbers
3. Mersenne numbers 101
3.1 History of the question . . . . . . . . . . . . . . . . . 101
3.2 Elementary properties of Mersenne numbers . . . . . 109
3.3 Mersenne primes: Prime divisors of Mersenne
numbers . . . . . . . . . . . . . . . . . . . . . . . . . 118
3.4 Mersenne primes: Lucas-Lehmer test . . . . . . . . . 123
3.5 Mersenne numbers in the family of special
numbers . . . . . . . . . . . . . . . . . . . . . . . . . 131
3.6 Open problems . . . . . . . . . . . . . . . . . . . . . 141
4. Fermat numbers 149
4.1 History of the question . . . . . . . . . . . . . . . . . 149
4.2 Elementary properties of Fermat numbers . . . . . . 156
4.3 Fermat primes: Prime divisors of Fermat
numbers . . . . . . . . . . . . . . . . . . . . . . . . . 170
4.4 Fermat primes: P´epin’s test . . . . . . . . . . . . . . 176
4.5 Fermat numbers in the family of special
numbers . . . . . . . . . . . . . . . . . . . . . . . . . 181
4.6 Open problems . . . . . . . . . . . . . . . . . . . . . 190
5. Modern Applications 195
5.1 On place of prime numbers in Mathematics . . . . . 195
5.2 Problems in Number Theory, connected with
Mersenne numbers . . . . . . . . . . . . . . . . . . . 202
5.3 Problems in Number Theory, connected with
Fermat numbers. . . . . . . . . . . . . . . . . . . . . 211
5.4 Prime numbers records and Mersenne numbers . . . 218
5.5 Mersenne and Fermat numbers in Cryptography . . 224
5.6 Open problems . . . . . . . . . . . . . . . . . . . . . 240
6. Zoo of Numbers 249
7. Mini Dictionary 259
8. Exercises 267
Bibliography 295
Index 303
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Notations
• X — a set; X = {x ,...,x } — a finite set; |X| = n — the number
1 n
n of elements in the finite set X = {x ,...,x }.
1 n
• N = {1,2,3,...} —thesetofpositive integers(ornatural numbers).
• Z = {...,−3,−2,−1,0,1,2,3,...} — the set of integers.
• Q — the set of rational numbers.
• R — the set of real numbers.
• C = {a+b·i :a,b ∈ R,i2 = −1} — the set of complex numbers.
• b|a — a non-zero integer b divides an integer a: a = bc, where
c ∈ Z.
• gcd(a ,...,a ) — greatest common divisor of integers a ,...,a , at
1 n 1 n
least one of which is non-equal to 0, i.e., the greatest integer,
dividing a ,...,a . If gcd(a ,...,a ) = 1, the numbers a ,...,a are
1 n 1 n 1 n
called relatively prime, or coprime; if gcd(a ,a ) = 1 for any dis-
i j
tinct i,j ∈ {1,...,n}, the numbers a ,...,a are called pairwise
1 n
relatively prime.
• lcm(a ,...,a ) — least common multiple of non-zero integers
1 n
a ,...,a , i.e., the least positive integer divided by a ,...,a .
1 n 1 n
• rest(a,b) — the remainder of an integer a after its integer division
byapositiveintegerb: a = b·q+rest(a,b),whereq,rest(a,b) ∈Z,
and 0 ≤ rest(a,b)< b.
ix
