Research Catalog

Details

Description

1. xxii, 362 pages : illustrations (some color); 24 cm

Series statement

1. Springer series in information sciences, 0720-678X ; 7

Uniform title

1. Springer series in information sciences ; 7.

Alternative title

1. Number theory applications in cryptography, physics, digital information, computing, and self-similarity

Subject

1. Number theory
2. Number theory
3. Anwendung
4. Zahlentheorie
5. Getaltheorie
6. Théorie des nombres

Contents

1. A few fundamentals : Introduction : Fibonacci, continued fractions and the golden ratio ; Fermat, primes and cyclotomy ; Euler, totients and cryptography ; Gauss, congruences and diffraction ; Galois, fields and codes -- The natural numbers : The fundamental theorem ; The least common multiple ; Planetary "gears" ; The greatest common divisor ; Human pitch perception ; Octaves, temperament, kilos and decibels ; Coprimes ; Euclid's algorithm -- Primes : How many primes are there? ; The sieve of Eratosthenes ; A Chinese theorem in error ; A formula for primes ; Mersenne primes ; Repunits ; Perfect numbers ; Fermat primes ; Gauss and the impossible heptagon -- The prime distribution : A probabilistic argument ; The prime-counting function ; David Hilbert and large nuclei ; Coprime probabilities ; Primes in progressions ; Primeless expansions ; Squarefree and coprime integers ; Twin primes ; Prime triplets ; Prime quadruplets and quintuplets ; Primes at any distance ; Spacing distribution between adjacent primes ; Goldbach's conjecture ; Sum of three primes -- Some simple applications : Fractions: continued, Egyptian and Farey : A neglected subject ; Relations with measure theory ; Periodic continued fractions ; Electrical networks and squared squares ; Fibonacci numbers and the golden ration ; Fibonacci, rabbits and computers ; Fibonacci and divisibility ; Generalized Fibonacci and Lucas numbers ; Egyptian fractions, inheritance and some unsolved problems ; Farey fractions : Farey trees ; Locked Pallas. Fibonacci and the problem of bank deposits ; Error-free computing -- Congruences and the like : Linear congruences : Residues ; Some simple fields ; Powers and congruences --
2. Diophantine equations : Relation with congruences ; A Gaussian trick ; Nonlinear diophantine equations ; Triangular numbers ; Pythagorean numbers ; Exponential Diophantine equations ; Fermat's last "theorem" ; The demise of a conjecture by Euler ; A nonlinear Diophantine equation in physics and the geometry of numbers ; Normal-mode degeneracy in room acoustics (a number-theoretic application) ; Warring's problem -- The theorems of Fermat, Wilson and Euler : Fermat's theorem ; Wilson's theorem ; Euler's theorem ; The impossible star of David ; Dirichlet and linear progression -- Cryptography and divisors : Euler trap doors and public-key encryption : A numerical trap door ; Digital encryption ; Public-key encryption ; A simple example ; Repeated encryption ; Summary and encryption requirements -- The divisor functions : The number of divisors ; The average of the divisor function ; The geometric mean of the divisors ; The summatory function of the divisor function ; The generalized divisor functions ; The average value of Euler's function -- The prime divisor functions : The number of different prime divisors ; The distribution of x(x) ; The number of prime divisors ; The harmonic mean of x(x) ; Medians and percentiles of x(x) ; Implications for public-key encryption -- Certified signatures : A story of creative financing ; Certified signature for public-key encryption -- Primitive roots : Orders ; Periods of decimal and binary fractions ; A primitive proof of Wilson's theorem ; The index--a number of theoretic logarithm ; Solution of exponential congruences ; What is the order of Tm of an integer m modulo a prime p? ; Index "encryption" ; A Fourier property of primitive and concert hall acoustics ; More spacious-sounding sound ; Galois arrays for z-ray astronomy ; A negative property of the Fermat primes -- Knapsack encryption : An easy knapsack ; A hard knapsack --
3. Residues and diffraction : Quadratic residues : Euler's criterion ; The Legendre symbol ; A Fourier property of Legendre sequences ; Gauss sums ; Pretty diffraction ; Quadratic reciprocity ; A Fourier property of quadratic-residue sequences ; Spread spectrum communication ; Generalized Legendre sequences obtained through complexification of the Euler criterion -- Chinese and other fast algorithms : The Chinese remainder theorem and simultaneous congruences : Simultaneous congruences ; The Sino-representation: a Chinese number system ; Applications of the Sino-representation ; Discrete Fourier transformation in Sino ; A Sino-optical Fourier transformer ; Generalized Sino-representation ; Fast prime-length Fourier transform -- Fast transformation and Kronecker products : A fast Hadamard transform ; The basic principle of the fast Fourier transforms -- Quadratic congruences : Application of the Chinese Remainder Theorem (CRT) -- Pseudoprimes, Mobius transform, and partitions : Pseudoprimes, poker and remote coin tossing : Pulling roots to ferret out composites ; Factors from a square root ; Coin tossing by telephone ; Absolute and strong pseudoprimes ; Fermat and strong pseudoprimes ; Deterministic primality testing ; A very simple factoring algorithm ; Factoring with elliptic curves ; Quantum factoring -- The Mobius function and the Mobius transform : The Mobius transform and its inverse ; Proof of the inversion formula ; Second inversion formula ; Third inversion formula ; Fourth inversion formula ; Riemann's hypothesis and the disproof of the Mertens conjuncture ; Dirichlet series and the Mobius function -- Generating functions and partitions : Generating functions ; Partitions of functions ; Generating functions of partitions ; Restricted partitions --
4. Cyclotomy and polynomials : Cyclotomic polynomials : How to divide a circle into equal parts ; Gauss's great insight ; Factoring in different fields ; Cyclotomy in the complex plane ; How to divide a circle with compass and straightedge : Rational factors of xx - 1. An alternative rational factorization ; Relation between rational factors and complex roots ; How to calculate with cyclotomic polynomials -- Linear systems and polynomials : Impulse responses ; Time-discrete systems and the z transform ; Discrete convolution ; Cyclotomic polynomials and z transform -- Polynomial theory : Some basic facts of polynomial life ; Polynomial residues ; Chinese remainders for polynomials ; Euclid's algorithm for polynomials -- Galois fields and more applications : Galois fields : Prime order ; Prime power order ; Generation of GF(2x) ; How many primitive elements? ; Recursive relations ; How to calculate in GF(px) ; Zech logarithm, doppler radar and optimum ambiguity functions ; A unique phase-array based on the Zech logarithm ; Spread-spectrum communication and Zech logarithms -- Spectral properties of Galois sequences : Circular correlation ; Application to error-correcting codes and speech recognition ; Application to precision measurements ; Concert hall measurements ; The fourth effect of general relativity ; Toward better concert hall acoustics ; Higher-dimensional diffusors ; Active array applications -- Random number generators : Pseudorandom Galois sequences ; Randomness from congruences ; "Continuous" distributions ; Four ways to generate Gaussian variable ; Pseudorandom sequences in cryptography -- Waveforms and radiation patterns : Special phases ; The Rudin-Shapiro polynomials ; Gauss sums and peak factors ; Galois sequences and the smallest peak factors ; Minimum redundancy antennas ; Golomb rulers -- Number theory, randomness and "art" : Number theory and graphic design ; The primes of Gauss and Eisenstein ; Galois fields and impossible necklaces ; "Baroque" integers -- Self-similarity, fractals and art : Self-similarity, fractals, deterministic chaos and a new state of matter : Fibonacci, noble numbers and a new state of matter ; Cantor sets, fractals and a musical paradox ; The twin dragon: a fractal from a complex number system ; Statistical fractals ; Some crazy mappings ; The logistic parabola and strange attractors ; Conclusion.

Owning institution

1. Princeton University Library

Bibliography (note)

1. Includes bibliographical references (p. [339]-349) and indexes.