Elementary Theory of Numbers

W. SIERPINSKI

Editor: A. SCHINZEL Mathematical Institute of the Polish Academy of Sciences

1988 NORTH-HOLLAND AMSTERDAM • NEW YORK • OXFORD

PWN-POLISH SCIENTIFIC PUBLISHERS WARSZAWA

CONTENTS

Author's Preface v Editor's Preface vii

CHAPTER I. DIVISIBILITY AND INDETERMINATE EQUATIONS OF FIRST DEGREE 1

1. Divisibility 1 2. Least common multiple 4 3. Greatest common divisor 5 4. Relatively prime numbers 6 5. Relation between the greatest common divisor and the least common

multiple 8 6. Fundamental theorem of arithmetic 9 7. Proof of the formulae (a1,a2,-..,a„+1) = ((alta2,...,a„),a„+1) and

[a1,a2>...,a„+1] = [[>i,a2 «J ,a„ + 1 ] 13 8. Rules.for calculating the greatest common divisor of two numbers 15 9. Representation of rationals as simple continued fractions 19

10. Linear form of the greatest common divisor 20 11. Indeterminate equations of m variables and degree 1 23 12. Chinese Remainder Theorem 28 13. Thue's Theorem 30 14. Square-free numbers 31

CHAPTER II. DIOPHANTINEANALYSIS OFSECOND AND HIGHER DEGREES 32

1. Diophantine equations of arbitrary degree and one unknown 32 2. Problems concerning Diophantine equations of two or more unknowns 33 3. The equation x2+y2 = z2 35 4. Integral Solutions of the equation x2+y2 = z2 for which x—y = ±1 42 5. Pythagorean triangles of the same area 46 6. On Squares whose sum and difference are Squares 50 7. The equation x*+y* = z2 57 8. On three Squares for which the sum of any two is a square 60 9. Congruent numbers 62

10. The equation x2+y2 + z2 = t2 66 11. The equation xy = zt 69 12. The equation x 4 - x V + / = z2 73 13. The equation x*+9x2y2+ 21y* = z2 75 14. The equation x3+y3 = 2z3 77 15. The equation x3+y3 = az3 with a > 2 82

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16. Triangulär numbers 84 17. The equation x2-Dy2 = 1 88 18. The equations x2 + k = y3 where k is an integer 101 19. On some exponential equations and others 109

CHAPTER III. PRIME NUMBERS 113

1. The primes. Factorization of a natural number m into primes 113 2. The Eratosthenes sieve. Tables of prime numbers 117 3. The differences between consecutive prime numbers 119 4. Goldbach's conjecture 123 5. Arithmetical progressions whose terms are prime numbers 126 6. Primes in a given arithmetical progression 128 7. Trinomial of Euler x2 + x + 41 130 8. The Conjecture H 133 9. The function n (x) 136

10. Proof of Bertrand's Postulate (Theorem of Tchebycheff) 137 11. Theorem of H. F. Scherk 148 12. Theorem of H.-E. Richert 151 13. A conjecture on prime numbers 153 14. Inequalities for the function n(x) 157 15. The prime number theorem and its consequences 162

CHAPTER IV. NUMBER OF DIVISORS AND THEIR SUM 166

1. Number of divisors 166 2. Sums d(l)+d(2) + ... +d(n) 169 3. Numbers d(n) as coefficients of expansions 173 4. Sum of divisors 174 5. Perfect numbers 182 6. Amicable numbers 186 7. The sum <7(l) + <r(2)+ ... +c(n) 188 8. The numbers a(n) as coefficients of various expansions 190 9. Sums of summands depending on the natural divisors of a natural number n 191

10. The Möbius function 192 11. The Liouville function k(n) 196

CHAPTER V. CONGRUENCES 198

1. Congruences and their simplest properties 198 2. Roots of congruences. Complete set of residues 203 3. Roots of polynomials and roots of congruences 206 4. Congruences of the first degree 209 5. Wilson's theorem and the simple theorem of Fermat 211 6. Numeri idonei 228 7. Pseudoprime and absolutely pseudoprime numbers 229

CONTENTS XI

8. Lagrange's theorem 235 9. Congruences of the second degree 239

CHAPTER VI. EULER'S TOTIENT FUNCTION AND THE THEOREM OF EULER 245

1. Euler's totient function 245 2. Properties of Euler's totient function 257 3. The theorem of Euler 260 4. Numbers which belong to a given exponent with respect to a given

modulus 263 5. Proof of the existence of infinitely many primes in the arithmetical

Progression nk +1 268 6. Proof of the existence of the primitive root of a prime number 272 7. An nth power residue for a prime modulus p 276 8. Indices, their properties and applications .279

CHAPTER VII. R EPRESENTATION OF NUMBERS BY DECIMALS IN A GIVEN SCALE 285

1. Representation of natural numbers by decimals in a given scale 285 2. Representation« of numbers by decimals in negative scales 290 3. Infinite fractions in a given scale 291 4. Representations of rational numbers by decimals 295 5. Normal numbers and absolutely normal numbers 299 6. Decimals in the varying scale 300

CHAPTER VIII. CONTINUED FRACTIONS 304

1. Continued fractions and their convergents 304 2. Representation of irrational numbers by continued fractions 306 3. Law of the best approximation 312 4. Continued fractions of quadratic irrationals _ 313 5. Application of the continued fraction for ^/D in solving the equations

x2-Dy2 = 1 and x2-Dy2 = - 1 329 6. Continued fractions other than simple continued fractions 335

CHAPTERIX. LEGENDRES SYMBOL AND JACOBIS SYMBOL 340

1. Legendre's symbol ( — 1 and its properties 340

2. The quadratic reciprocity law 346 3. Calculation of Legendre's symbol by its properties 351 4. Jacobi's symbol and its properties 352 5. Eisenstein's rule 355

CHAPTER X. MERSENNE NUMBERS AND FERMAT NUMBERS 360

1. Some properties of Mersenne numbers 360 2. Theorem of E. Lucas and D. H. Lehmer 363

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3. How the greatest of the known prime numbers have been found 367 4. Prime divisors of Fermat numbers 369 5. A necessary and sufficient condition for a Fermat number to be a prime 375

CHAPTER XI. REPRESENTATIONS OF NATURAL NUMBERS AS SUMS OF NON-

NEGATIVE feth POWERS 378

1. Sums of two Squares 378 2. The average number of representations as sums of two Squares 381 3. Sums of two Squares of natural numbers 388 4. Sums of three Squares 391 5. Representation by four Squares 397 6. The sums of the Squares of four natural numbers 402 7. Sums of m > 5 positive Squares 408 8. The difference of two Squares 410 9. Sums of two cubes 412

10. The equation x3+y3 = z3 415 11. Sums of three cubes 419 12. Sums of four cubes 422 13. Equal sums of different cubes 424 14. Sums of biquadrates 425 15. Waring's theorem 427

CHAPTERXII. SOME PROBLEMS OF THE ADDITIVE THEORYOF NUMBERS 431

1. Partitio numerorum 431 2. Representations as sums of n non-negative summands 433 3. Magic Squares 434 4. Schur's theorem and its corollaries 439 5. Odd numbers which are not of the form 2k+p, where p is a prime 445

CHAPTER XIII. COMPLEX INTEGERS 449

1. Complex integers and their norm. Associated integers 449 2. Euclidean algorithm and the greatest common divisor of complex integers 453 3. The least common multiple of complex integers 458 4. Complex primes 459 5. The factorization of complex integers into complex prime factors 463 6. The number of complex integers with a given norm 465 7. Jacobi's four-square theorem 469

Bibliography 482

Author index 505

Subject index 511