Elementary Number Theory

399 Pages·2007·1.31 MB·english

by  Peter Hackman

Elementary Number Theory
Peter Hackman
HHH Productions
November 5, 2007
ii
(cid:13)c P Hackman, 2007.
Contents
Preface ix
A Divisibility, Unique Factorization 1
A.I The gcd and B´ezout . . . . . . . . . . . . . . . . . . . . . . 1
A.II Two Divisibility Theorems . . . . . . . . . . . . . . . . . . 6
A.III Unique Factorization . . . . . . . . . . . . . . . . . . . . . 8
A.IV Residue Classes, Congruences . . . . . . . . . . . . . . . . . 11
A.V Order, Little Fermat, Euler . . . . . . . . . . . . . . . . . . 20
A.VI A Brief Account of RSA . . . . . . . . . . . . . . . . . . . . 32
B Congruences. The CRT. 35
B.I The Chinese Remainder Theorem . . . . . . . . . . . . . . 35
B.II Euler’s Phi Function Revisited . . . . . . . . . . . . . . . . 42
* B.III General CRT . . . . . . . . . . . . . . . . . . . . . . . . . . 46
B.IV Application to Algebraic Congruences . . . . . . . . . . . . 51
B.V Linear Congruences . . . . . . . . . . . . . . . . . . . . . . 52
B.VI Congruences Modulo a Prime . . . . . . . . . . . . . . . . . 54
B.VII Modulo a Prime Power . . . . . . . . . . . . . . . . . . . . 58
C Primitive Roots 67
iii
iv CONTENTS
C.I False Cases Excluded . . . . . . . . . . . . . . . . . . . . . 67
C.II Primitive Roots Modulo a Prime . . . . . . . . . . . . . . . 70
C.III Binomial Congruences . . . . . . . . . . . . . . . . . . . . . 73
C.IV Prime Powers . . . . . . . . . . . . . . . . . . . . . . . . . . 78
C.V The Carmichael Exponent . . . . . . . . . . . . . . . . . . . 85
* C.VI Pseudorandom Sequences . . . . . . . . . . . . . . . . . . . 89
C.VII Discrete Logarithms . . . . . . . . . . . . . . . . . . . . . . 91
* C.VIII Computing Discrete Logarithms . . . . . . . . . . . . . . . 92
D Quadratic Reciprocity 103
D.I The Legendre Symbol . . . . . . . . . . . . . . . . . . . . . 103
D.II The Jacobi Symbol . . . . . . . . . . . . . . . . . . . . . . 114
D.III A Cryptographic Application . . . . . . . . . . . . . . . . . 119
D.IV Gauß’ Lemma . . . . . . . . . . . . . . . . . . . . . . . . . 119
D.V The “Rectangle Proof” . . . . . . . . . . . . . . . . . . . . 123
D.VI Gerstenhaber’s Proof . . . . . . . . . . . . . . . . . . . . . 125
* D.VII Zolotareff’s Proof . . . . . . . . . . . . . . . . . . . . . . . 127
E Some Diophantine Problems 139
E.I Primes as Sums of Squares . . . . . . . . . . . . . . . . . . 139
E.II Composite Numbers . . . . . . . . . . . . . . . . . . . . . . 146
E.III Another Diophantine Problem . . . . . . . . . . . . . . . . 152
E.IV Modular Square Roots . . . . . . . . . . . . . . . . . . . . . 156
E.V Applications . . . . . . . . . . . . . . . . . . . . . . . . . . 161
F Multiplicative Functions 163
F.I Definitions and Examples . . . . . . . . . . . . . . . . . . . 163
CONTENTS v
F.II The Dirichlet Product . . . . . . . . . . . . . . . . . . . . . 165
F.III M¨obius Inversion . . . . . . . . . . . . . . . . . . . . . . . . 168
* F.IV Two More Results . . . . . . . . . . . . . . . . . . . . . . . 171
F.V Primitive Roots, Again . . . . . . . . . . . . . . . . . . . . 175
F.VI A Combinatorial Application . . . . . . . . . . . . . . . . . 177
F.VII The Sum of Divisors . . . . . . . . . . . . . . . . . . . . . . 180
F.VIII Cyclotomic Polynomials . . . . . . . . . . . . . . . . . . . . 183
G Continued Fractions 191
G.I Motive, Definitions . . . . . . . . . . . . . . . . . . . . . . . 191
G.II Basic Theorems . . . . . . . . . . . . . . . . . . . . . . . . 195
* G.III Negative Irrationalities . . . . . . . . . . . . . . . . . . . . 201
G.IV Best Rational Approximation . . . . . . . . . . . . . . . . . 202
G.V Where Are the “Continued Fractions”? . . . . . . . . . . . 206
G.VI Finite Continued Fractions . . . . . . . . . . . . . . . . . . 209
G.VII Cornacchia’s Algorithm . . . . . . . . . . . . . . . . . . . . 212
H “QCF” and Pell’s Equation 219
H.I An Algorithm for Quadratic Irrationalities . . . . . . . . . . 219
H.II Conjugates . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
H.III x2 −Dy2 = ±1 . . . . . . . . . . . . . . . . . . . . . . . . . 224
H.IV x2 −Dy2 = N . . . . . . . . . . . . . . . . . . . . . . . . . 236
H.V Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . 240
H.VI Periodicity . . . . . . . . . . . . . . . . . . . . . . . . . . . 241
* H.VII Periodicity, Continued . . . . . . . . . . . . . . . . . . . . . 242
J Special Topics 251
vi CONTENTS
J.I Matrix Notation . . . . . . . . . . . . . . . . . . . . . . . . 251
J.II Equivalent Quadratic Irrationalities . . . . . . . . . . . . . 253
J.III Equivalence and Continued Fractions . . . . . . . . . . . . 254
* J.IV An Alternative Approach . . . . . . . . . . . . . . . . . . . 259
J.V Reciprocal Expansions . . . . . . . . . . . . . . . . . . . . . 262
J.VI Selfreciprocity, Partial Quotients . . . . . . . . . . . . . . . 264
J.VII Symmetry of the P ,Q . . . . . . . . . . . . . . . . . . . . 266
k k
* J.VIII Moving Forward One Period . . . . . . . . . . . . . . . . . 267
* J.IX Running a Period Backwards . . . . . . . . . . . . . . . . . 271
* J.X More on Pell-Like Equations . . . . . . . . . . . . . . . . . 273
J.XI More on x2 −Dy2 = N . . . . . . . . . . . . . . . . . . . . 279
K Z[i], Other Number Rings 287
K.I Preparations . . . . . . . . . . . . . . . . . . . . . . . . . . 287
K.II Unique Factorization in Z[i] . . . . . . . . . . . . . . . . . . 292
K.III The Number of Representations . . . . . . . . . . . . . . . 301
√
K.IV Z[−1 +i 3] . . . . . . . . . . . . . . . . . . . . . . . . . . 303
2 2
L Primality and Factorization 309
L.I Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 309
L.II Special Numbers, Special Factors . . . . . . . . . . . . . . . 310
L.III Trial Division . . . . . . . . . . . . . . . . . . . . . . . . . . 313
L.IV Lists of Primes . . . . . . . . . . . . . . . . . . . . . . . . . 313
L.V Fast Exponentiation . . . . . . . . . . . . . . . . . . . . . . 316
L.VI Primality: Fermat and Miller-Rabin . . . . . . . . . . . . . 317
L.VII Lehmer, Lucas, Pocklington . . . . . . . . . . . . . . . . . . 323
L.VIII Factoring: Pollard p−1 . . . . . . . . . . . . . . . . . . . . 331
CONTENTS vii
L.IX Factoring, Pollard rho . . . . . . . . . . . . . . . . . . . . . 332
L.X “Waiting For a Square” . . . . . . . . . . . . . . . . . . . . 340
L.XI CFRAC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343
L.XII Elimination Modulo 2 . . . . . . . . . . . . . . . . . . . . . 350
L.XIII Quadratic Sieve, Orientation . . . . . . . . . . . . . . . . . 354
L.XIV Lucas Sequences and Primality . . . . . . . . . . . . . . . . 358
L.XV Mersenne Numbers . . . . . . . . . . . . . . . . . . . . . . 367
L.XVI Lucas and Modular Square Roots . . . . . . . . . . . . . . 370
L.XVII Scalar Formulas . . . . . . . . . . . . . . . . . . . . . . . . 373
Bibliography 377
Tables 385
viii CONTENTS
Preface
My last new assignment at Link¨oping University was to a course in Elemen-
tary Number Theory. While somewhat removed from my algebraic interests
and competence, that course (which I conducted for five years) turned out
to be an extremely rewarding and challenging experience, largely due to the
enthusiasm of the students.
Never content to just copy text books, I put a lot of energy into finding
new proofs, new ways of organizing the material, and some at least unusual
topics. This quest for inspiration produced a lot of notes that I decided to
compile into a short text on retiring from the University. As you might have
guessed, the project grew into a full-size book.
What I hoped to contribute to the existing literature is a perhaps slimmer
and more affordable volume. At the same time I wished to include some
exciting and challenging, yet completely elementary, material not found in
current texts.
“Elementary” means that almost no Analysis is used, and almost no “Ab-
stract” Algebra. Algebra really becomes abstract only with the introduction
of techniques like homomorphisms, direct sums and quotient constructions.
We do, however, speak of (number) rings, fields, and residue classes of inte-
gers, and their arithemetic.
Among the more unusual material is a reasonably complete account of Cor-
nacchia’s algorithm for solving x2 +Dy2 = p, using Euclid’s Algorithm, and
that of Lagrange (revived by Matthews and Mollin) for x2 −Dy2 = N, us-
ing infinite continued fractions. There are strong analogies between the two
theories, which I emphasize by using exactly the same wording in several
parallel passages. Modular square roots is another of my favorite topics,
and I present two algorithms for them, that of Berlekamp and the one using
Lucas sequences (or, equivalently, Cipolla’s algorithm) each exploiting some
interesting theoretical item.
ix
x PREFACE
To keep the material within bounds I had to make some clear decisions what
not to include. One main decision was to not give proofs of the statistics and
complexity of algorithms. There are comprehensive accounts in the books by
Riesel, Bach-Shallit, and Crandall-Pomerance cited in the Bibliography.
Another decision was to give fewer applications. The motivational value
of applications does not depend on their quantity; besides, who wants to
compete with such excellent texts (in Cryptography) as Trappe-Washington,
or Buchmann? To be sure, almost all of the math relevant to these accounts,
is included here. Whenever I found an application worthy of inclusion it was
usually because it strengthened some of the main theoretical ideas of the
text.
A few words about style. For the most part I stick to the strict Defintion-
Example-Theorem-Proof-Example format simply because I want to make it
clear where things begin and end. Also, many readers (like me) will want to
skip longer proofs on a first reading.
Having dabbled in journalism I try to paragraph and display often. I have
alsotriedtominimizethenumberofcross-references. Further, bynumbering
everything in one sequence, and boxing theorems and definitions I hope to
make it easier on the reader. As I never refer back to a subsection, these are
un-numbered.
Finally, there is no numbering of equations, as reference to them is strictly
local. They are labeled by one to three stars. I simply hate references like
“we now return to the study of the differential operator (17) in Section XVII
of Chapter Q” (that operator of course being the Laplacian).
Books are referred to by author name. Very few articles are cited; in these
cases I give full reference in the text, but I have also collected them at the
end of the text.
I strongly advocate the use of computers as a means of generating and inves-
tigating examples. To really understand an algorithm, or a result, it helps
to program it. There are suggestions for computer projects, ranging in com-
plexity from a few lines to maybe one or two pages.
Hopefully the suggested projects still allow the math to dominate over more
delicate programming issues. Ideally much more work should be spent on
checking and tracing, and varying the input and the parameters, than on de-
vising the program. This text is not conceived as a book on “computational”
number theory.