.

5 (hodnocen1 x )

BK

Zagreb : Školska knjiga, 2021

x, 621 stran ; 25 cm

ISBN 978-953-0-30897-8 (vázáno)

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Preface to the Croatian edition i // Preface to the English edition v // 1 Introduction 1 // 1.1 Peano’s axioms 1 // 1.2 Principle of mathematical induction 4 // 1.3 Fibonacci numbers 10 // 1.4 Exercises 18 // 2 Divisibility 22 // 2.1 Greatest common divisor 22 // 2.2 Euclid’s algorithm 25 // 2.3 Prime numbers 31 // 2.4 Exercises 39 // 3 Congruences 42 // 3.1 Definition and properties of congruences 42 // 3.2 Tests of divisibility 45 // 3.3 Linear congruences 48 // 3.4 Chinese remainder theorem 50 // 3.5 Reduced residue system 54 // 3.6 Congruences with a prime modulus 57 // 3.7 Primitive roots and indices 62 // 3.8 Representations of rational numbers by decimals 68 // 3.9 Pseudoprimes 73 // 3.10 Exercises 79 // 4 Quadratic residues 83 // 4.1 Legendre symbol 83 // 4.2 Law of quadratic reciprocity 89 // • O // VII // Vlll // 4.3 Computing square roots modulo p 94 // 4.4 Jacobi symbol 96 // 4.5 Divisibility of Fibonacci numbers 99 // 4.6 Exercises 104 // 5 Quadratic forms 107 // 5.1 Sums of two squares 107 // 5.2 Positive definite binary quadratic forms Ill // 5.3 Sums of four squares 121 // 5.4 Sums of three squares 125 // 5.5 Exercises 132 // 6 Arithmetical functions 136 // 6.1 Greatest integer function 136 // 6.2 Multiplicative functions 140 // 6.3 Asymptotic estimates for arithmetical functions 145 // 6.4 Dirichlet product 152 // 6.5 Exercises 155 // 7 Distribution of primes 159 // 7.1 Elementary estimates for the function T(x) 159 // 7.2 Chebyshev functions 164 // 7.3 The Riemann zeta function 172 // 7.4 Dirichlet characters 176 // 7.5 Primes in arithmetic progressions 183 // 7.6 Exercises 187 // 8 Diophantine approximation 191 // 8.1 Dirichlet’s theorem 191 // 8.2 Farey sequences 194 // 8.3 Continued fractions 201 // 8.4 Continued fraction and approximations to irrational numbers 208 // 8.5 Equivalent numbers 217 // 8.6 Periodic continued fractions 222 //

8.7 Newton’s approximants 229 // 8.8 Simultaneous approximations 233 // 8.9 LLL algorithm 240 // 8.10 Exercises 246 // 9 Applications of Diophantine approximation to cryptography 250 // 9.1 A very short introduction to cryptography 250 // 9.2 RSA cryptosystem 254 // 9.3 Wiener’s attack on RSA 257 // 9.4 Attacks on RSA using the LLL algorithm 260 // 9.5 Coppersmith’s theorem 264 // 9.6 Exercises 267 // 10 Diophantine equations 1 270 // 10.1 Linear Diophantine equations 270 // 10.2 Pythagorean triangles 274 // 10.3 Pell’s equation 284 // 10.4 Continued fractions and Pell’s equation 293 // 10.5 Pellian equation 296 // 10.6 Squares in the Fibonacci sequence 302 // 10.7 Ternary quadratic forms 307 // 10.8 Local-global principle 320 // 10.9 Exercises 328 // 11 Polynomials // 11.1 Divisibility of polynomials // 11.2 Polynomial roots // 11.3 Irreducibility of polynomials // 11.4 Polynomial decomposition // 11.5 Symmetric polynomials // 11.6 Exercises // 12 Algebraic numbers 366 // 12.1 Quadratic fields 366 // 12.2 Algebraic number fields 376 // 12.3 Algebraic integers 380 // 12.4 Ideals 384 // 12.5 Units and ideal classes 392 // 12.6 Exercises 399 // 13 Approximation of algebraic numbers 402 // 13.1 Liouville’s theorem 402 // 13.2 Roth’s theorem 404 // 13.3 The hypergeometric method 407 // 13.4 Approximation by quadratic irrationals 417 // X // 13.5 Polynomial root separation 422 // 13.6 Exercises 428 // 14 Diophantine equations II 431 // 14.1 Thue equations 431 // 14.2 Tzanakis’ method 435 // 14.3 Linear forms in logarithms 440 // 14.4 Baker-Davenport reduction 445 // 14.5 LLL reduction 450 // 14.6 Diophantine m-tuples 454 // 14.7 Exercises 462 // 15 Elliptic curves 466 // 15.1 Introduction to elliptic curves 466 // 15.2 Equations of elliptic curves 473 // 15.3 Torsion group 486 // 15.4 Canonical height and Mordell-Weil theorem 499 // 15.5 Rank of elliptic curves 506 //

15.6 Finite fields 519 // 15.7 Elliptic curves over finite fields 526 // 15.8 Applications of elliptic curves in cryptography 535 // 15.9 Primality proving using elliptic curves 544 // 15.10 Elliptic curve factorization method 548 // 15.11 Exercises 552 // 16 Diophantine problems and elliptic curves 556 // 16.1 Congruent numbers 556 // 16.2 Mordell’s equation 558 // 16.3 Applications of factorization in quadratic fields 560 // 16.4 Transformation of elliptic curves to Thue equations 565 // 16.5 Algorithm for solving Thue equations 568 // 16.6 abc conjecture 574 // 16.7 Diophantine m-tuples and elliptic curves 578 // 16.8 Exercises 586 // References 589 // Notation Index 613 // Subject Index 616