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Number Theory Revealed: A Masterclass [Hardback]

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  • Formāts: Hardback, 607 pages, height x width: 254x178 mm
  • Izdošanas datums: 01-Dec-2019
  • Izdevniecība: American Mathematical Society
  • ISBN-10: 1470441586
  • ISBN-13: 9781470441586
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Number Theory Revealed: A MasterclassPalielināt
  • Formāts: Hardback, 607 pages, height x width: 254x178 mm
  • Izdošanas datums: 01-Dec-2019
  • Izdevniecība: American Mathematical Society
  • ISBN-10: 1470441586
  • ISBN-13: 9781470441586
Citas grāmatas par šo tēmu:
Permanent link: https://www.kriso.lv/db/9781470441586.html
Keywords:
Number Theory Revealed: A Masterclass presents a fresh take on congruences, power residues, quadratic residues, primes, and Diophantine equations and presents hot topics like cryptography, factoring, and primality testing. Students are also introduced to beautiful enlightening questions like the structure of Pascal's triangle mod p, Fermat's Last Theorem for polynomials, and modern twists on traditional questions.

This Masterclass edition contains many additional chapters and appendices not found in Number Theory Revealed: An Introduction. It is ideal for instructors who wish to tailor a class to their own interests and gives well-prepared students further opportunities to challenge themselves and push beyond core number theory concepts, serving as a springboard to many current themes in mathematics. Additional topics in A Masterclass include the curvature of circles in a tiling of a circle by circles, the latest discoveries on gaps between primes, magic squares of primes, a new proof of Mordell's Theorem for congruent elliptic curves, as well as links with algebra, analysis, cryptography, and dynamics.
Preface xvii
Gauss's Disquisitiones Arithmeticae xxiii
Notation xxv
The language of mathematics xxvi
Prerequisites xxvii
Preliminary
Chapter on Induction		 1(8)
0.1 Fibonacci numbers and other recurrence sequences
1(2)
0.2 Formulas for sums of powers of integers
3(1)
0.3 The binomial theorem, Pascal's triangle, and the binomial coefficients
4(5)
Appendices for Preliminary
Chapter on Induction
0A A closed formula for sums of powers
9(2)
0B Generating functions
11(4)
0C Finding roots of polynomials
15(4)
0D What is a group?
19(3)
0E Rings and fields
22(3)
0F Symmetric polynomials
25(5)
0G Constructibility
30(3)
Chapter 1 The Euclidean algorithm
33(28)
1.1 Finding the gcd
33(2)
1.2 Linear combinations
35(2)
1.3 The set of linear combinations of two integers
37(2)
1.4 The least common multiple
39(1)
1.5 Continued fractions
39(2)
1.6 Tiling a rectangle with squares
41(4)
Appendices for
Chapter 1
1A Reformulating the Euclidean algorithm
45(6)
1B Computational aspects of the Euclidean algorithm
51(3)
1C Magic squares
54(3)
1D The Frobenius postage stamp problem
57(2)
1E Egyptian fractions
59(2)
Chapter 2 Congruences
61(20)
2.1 Basic congruences
1(63)
2.2 The trouble with division
64(2)
2.3 Congruences for polynomials
66(1)
2.4 Tests for divisibility
66(5)
Appendices for
Chapter 2
2A Congruences in the language of groups
71(4)
2B The Euclidean algorithm for polynomials
75(6)
Chapter 3 The basic algebra of number theory
81(46)
3.1 The Fundamental Theorem of Arithmetic
81(2)
3.2 Abstractions
83(2)
3.3 Divisors using factorizations
85(2)
3.4 Irrationality
87(1)
3.5 Dividing in congruences
88(2)
3.6 Linear equations in two unknowns
90(2)
3.7 Congruences to several moduli
92(2)
3.8 Square roots of 1 (mod n)
94(5)
Appendices for
Chapter 3
3A Factoring binomial coefficients and Pascal's triangle modulo p
99(5)
3B Solving linear congruences
104(5)
3C Groups and rings
109(3)
3D Unique factorization revisited
112(4)
3E Gauss's approach
116(1)
3F Fundamental theorems and factoring polynomials
117(6)
3G Open problems
123(4)
Chapter 4 Multiplicative functions
127(28)
4.1 Euler's φ-function
128(1)
4.2 Perfect numbers. "The whole is equal to the sum of its parts."
129(5)
Appendices for
Chapter 4
4A More multiplicative functions
134(6)
4B Dirichlet series and multiplicative functions
140(4)
4C Irreducible polynomials modulo p
144(3)
4D The harmonic sum and the divisor function
147(6)
4E Cyclotomic polynomials
153(2)
Chapter 5 The distribution of prime numbers
155(60)
5.1 Proofs that there are infinitely many primes
155(2)
5.2 Distinguishing primes
157(2)
5.3 Primes in certain arithmetic progressions
159(1)
5.4 How many primes are there up to xl
160(3)
5.5 Bounds on the number of primes
163(2)
5.6 Gaps between primes
165(2)
5.7 Formulas for primes
167(4)
Appendices for
Chapter 5
5A Bertrand's postulate and beyond
171(11)
Bonus read: A review of prime problems
175(1)
Prime values of polynomials in one variable
175(2)
Prime values of polynomials in several variables
177(2)
Goldbach's conjecture and variants
179(3)
5B An important proof of infinitely many primes
182(5)
5C What should be true about primes?
187(5)
5D Working with Riemann's zeta-function
192(6)
5E Prime patterns: Consequences of the Green-Tao Theorem
198(4)
5F A panoply of prime proofs
202(2)
5G Searching for primes and prime formulas
204(4)
5H Dynamical systems and infinitely many primes
208(7)
Chapter 6 Diophantine problems
215(20)
6.1 The Pythagorean equation
215(3)
6.2 No solutions to a Diophantine equation through descent
218(2)
6.3 Fermat's "infinite descent"
220(1)
6.4 Fermat's Last Theorem
221(4)
Appendices for
Chapter 6
6A Polynomial solutions of Diophantine equations
225(4)
6B No Pythagorean triangle of square area via Euclidean geometry
229(4)
6C Can a binomial coefficient be a square?
233(2)
Chapter 7 Power residues
235(60)
7.1 Generating the multiplicative group of residues
236(1)
7.2 Fermat's Little Theorem
237(3)
7.3 Special primes and orders
240(1)
7.4 Further observations
240(1)
7.5 The number of elements of a given order, and primitive roots
241(4)
7.6 Testing for composites, pseudoprimes, and Carmichael numbers
245(1)
7.7 Divisibility tests, again
246(1)
7.8 The decimal expansion of fractions
246(2)
7.9 Primes in arithmetic progressions, revisited
248(4)
Appendices for
Chapter 7
7A Card shuffling and Fermat's Little Theorem
252(6)
7B Orders and primitive roots
258(7)
7C Finding nth roots modulo prime powers
265(4)
7D Orders for finite groups
269(4)
7E Constructing finite fields
273(5)
7F Sophie Germain and Fermat's Last Theorem
278(2)
7G Primes of the form 2n + k
280(4)
7H Further congruences
284(6)
7I Primitive prime factors of recurrence sequences
290(5)
Chapter 8 Quadratic residues
295(42)
8.1 Squares modulo prime p
295(2)
8.2 The quadratic character of a residue
297(3)
8.3 The residue -1
300(1)
8.4 The residue 2
301(2)
8.5 The law of quadratic reciprocity
303(2)
8.6 Proof of the law of quadratic reciprocity
305(2)
8.7 The Jacobi symbol
307(2)
8.8 The squares modulo m
309(6)
Appendices for
Chapter 8
8A Eisenstein's proof of quadratic reciprocity
315(4)
8B Small quadratic non-residues
319(4)
8C The first proof of quadratic reciprocity
323(3)
8D Dirichlet characters and primes in arithmetic progressions
326(7)
8E Quadratic reciprocity and recurrence sequences
333(4)
Chapter 9 Quadratic equations
337(28)
9.1 Sums of two squares
337(3)
9.2 The values of x2 + dy2
340(1)
9.3 Is there a solution to a given quadratic equation?
341(3)
9.4 Representation of integers by ax2 + by2 with x, y rational, and beyond
344(1)
9.5 The failure of the local-global principle for quadratic equations in integers
345(1)
9.6 Primes represented by x2 + 5y2
345(3)
Appendices for
Chapter 9
9A Proof of the local-global principle for quadratic equations
348(5)
9B Reformulation of the local-global principle
353(3)
9C The number of representations
356(4)
9D Descent and the quadratics
360(5)
Chapter 10 Square roots and factoring
365(38)
10.1 Square roots modulo n
365(1)
10.2 Cryptosystems
366(2)
10.3 RSA
368(2)
10.4 Certificates and the complexity classes P and NP
370(2)
10.5 Polynomial time primality testing
372(1)
10.6 Factoring methods
373(3)
Appendices for
Chapter 10
10A Pseudoprime tests using square roots of 1
376(4)
10B Factoring with squares
380(3)
10C Identifying primes of a given size
383(4)
10D Carmichael numbers
387(4)
10E Cryptosystems based on discrete logarithms
391(2)
10F Running times of algorithms
393(2)
10G The AKS test
395(4)
10H Factoring algorithms for polynomials
399(4)
Chapter 11 Rational approximations to real numbers
403(40)
11.1 The pigeonhole principle
403(3)
11.2 Pell's equation
406(4)
11.3 Descent on solutions of x2 -- dy2 = n, d > 0
410(1)
11.4 Transcendental numbers
411(3)
11.5 The a&c-conjecture
414(4)
Appendices for
Chapter 11
11A Uniform distribution
418(5)
11B Continued fractions
423(15)
11C Two-variable quadratic equations
438(1)
11D Transcendental numbers
439(4)
Chapter 12 Binary quadratic forms
443(44)
12.1 Representation of integers by binary quadratic forms
444(2)
12.2 Equivalence classes of binary quadratic forms
446(1)
12.3 Congruence restrictions on the values of a binary quadratic form
447(1)
12.4 Class numbers
448(1)
12.5 Class number one
449(7)
Appendices for
Chapter 12
12A Composition rules: Gauss, Dirichlet, and Bhargava
456(9)
12B The class group
465(3)
12C Binary quadratic forms of positive discriminant
468(3)
12D Sums of three squares
471(4)
12E Sums of four squares
475(4)
12F Universality
479(3)
12G Integers represented in Apollonian circle packings
482(5)
Chapter 13 The anatomy of integers
487(14)
13.1 Rough estimates for the number of integers with a fixed number of prime factors
487(1)
13.2 The number of prime factors of a typical integer
488(3)
13.3 The multiplication table problem
491(1)
13.4 Hardy and Ramanujan's inequality
492(1)
Appendices for
Chapter 13
13A Other anatomies
493(4)
13B Dirichlet L-functions
497(4)
Chapter 14 Counting integral and rational points on curves, modulo p
501(14)
14.1 Diagonal quadratics
501(2)
14.2 Counting solutions to a quadratic equation and another proof of quadratic reciprocity
503(1)
14.3 Cubic equations modulo p
504(1)
14.4 The equation Eb: y2 = x3 + b
505(2)
14.5 The equation y2 -- x3 + ax
507(2)
14.6 A more general viewpoint on counting solutions modulo p
509(2)
Appendices for
Chapter 14
14A Gauss sums
511(4)
Chapter 15 Combinatorial number theory
515(20)
15.1 Partitions
515(2)
15.2 Jacobi's triple product identity
517(2)
15.3 The Freiman-Ruzsa Theorem
519(3)
15.4 Expansion and the Plunnecke-Ruzsa inequality
522(1)
15.5 Schnirel'man's Theorem
523(2)
15.6 Classical additive number theory
525(3)
15.7 Challenging problems
528(2)
Appendices for
Chapter 15
15A Summing sets modulo p
530(2)
15B Summing sets of integers
532(3)
Chapter 16 The p-adic numbers
535(12)
16.1 The p-adic norm
535(1)
16.2 p-adic expansions
536(1)
16.3 p-adic roots of polynomials
537(2)
16.4 p-adic factors of a polynomial
539(2)
16.5 Possible norms on the rationals
541(1)
16.6 Power series convergence and the p-adic logarithm
542(3)
16.7 The p-adic dilogarithm
545(2)
Chapter 17 Rational points on elliptic curves
547(38)
17.1 The group of rational points on an elliptic curve
548(3)
17.2 Congruent number curves
551(2)
17.3 No non-trivial rational points by descent
553(1)
17.4 The group of rational points of y2 = x3 -- x
553(1)
17.5 Mordell's Theorem: EA{Q) is finitely generated
554(4)
17.6 Some nice examples
558(3)
Appendices for
Chapter 17
17A General Mordell's Theorem
561(2)
17B Pythagorean triangles of area 6
563(2)
17C 2-parts of abelian groups
565(1)
17D Waring's problem
566(3)
Hints for exercises
569(14)
Recommended further reading
583(2)
Index 585
Andrew Granville is the Canada Research Chair in Number Theory at the University of Montreal and professor of mathematics at University College London. He has won several international writing prizes for exposition in mathematics, including the 2008 Chauvenet Prize and the 2019 Halmos-Ford Prize, and is the author of Prime Suspects (Princeton University Press, 2019), a beautifully illustrated graphic novel murder mystery that explores surprising connections between the anatomies of integers and of permutations.