Since the publication of the first edition of this work, considerable progress has been made in many of the questions examined. This edition has been updated and enlarged, and the bibliography has been revised. The variety of topics covered here includes divisibility, diophantine equations, prime numbers (especially Mersenne and Fermat primes), the basic arithmetic functions, congruences, the quadratic reciprocity law, expansion of real numbers into decimal fractions, decomposition of integers into sums of powers, some other problems of the additive theory of numbers and the theory of Gaussian integers.
Author(s): W. Sierpinski
Series: North-Holland Mathematical Library 31
Edition: 2nd/Englsh
Publisher: Elsevier Science Ltd
Year: 1988
Language: English
Pages: 527
Front Cover......Page 1
Elementary Theory of Numbers......Page 4
Copyright Page......Page 5
Contents......Page 10
Author's Preface......Page 6
Editor's Preface......Page 8
1. Divisibility......Page 14
2. Least common multiple......Page 17
3. Greatest common divisor......Page 18
4. Relatively prime numbers......Page 19
5. Relation between the greatest common divisor and the least common multiple......Page 21
6. Fundamental theorem of arithmetic......Page 22
7. Proof of the formulae (al, a2, ..., an+1) = (al,a2, ..., an),an+1) and [a1, a2, ...,an+1] = [[a1, a2, ...,an], an+1]......Page 26
8. Rules for calculating the greatest common divisor of two numbers......Page 28
9. Representation of rationals as simple continued fractions......Page 32
10. Linear form of the greatest common divisor......Page 33
11. Indeterminate equations of m variables and degree 1......Page 36
12. Chinese Remainder Theorem......Page 41
13. Thue's Theorem......Page 43
14. Square-free numbers......Page 44
1. Diophantine equations of arbitrary degree and one unknown......Page 45
2. Problems concerning Diophantine equations of two or more unknowns......Page 46
3. The equation x2 +y2 = z2......Page 48
4. Integral solutions of the equation x2+y2 = z2 for which x–y = ±1......Page 55
5. Pythagorean triangles of the same area......Page 59
6. On squares whose sum and difference are squares......Page 63
7. The equation x4 +y4 = z2......Page 70
8. On three squares for which the sum of any two is a square......Page 73
9. Congruent numbers......Page 75
10. The equation x2 + y2+ z2 = t2......Page 79
11. The equation xy = zt......Page 82
12. The equation x4 – x2y2 + y4 = z2......Page 86
13. The equation x4 + 9x2y2 +27y4 = z2......Page 88
14. The equation x3 + y3 = 2z3......Page 90
15. The equation x3 + y3 = az3 with a > 2......Page 95
16. Triangular numbers......Page 97
17. The equation x2 - Dy2 = 1......Page 101
18. The equations x2 +k = y3 where k is an integer......Page 114
19. On some exponential equations and others......Page 122
1. The primes. Factorization of a natural number m into primes......Page 126
2. The Eratosthenes sieve. Tables of prime numbers......Page 130
3. The differences between consecutive prime numbers......Page 132
4. Goldbach's conjecture......Page 136
5. Arithmetical progressions whose terms are prime numbers......Page 139
6. Primes in a given arithmetical progression......Page 141
7. Trinomial of Euler x2 + x+ 41......Page 143
8. The Conjecture H......Page 146
9. The function π (x)......Page 149
10. Proof of Bertrand's Postulate (Theorem of Tchebycheff)......Page 150
11. Theorem of H. F. Scherk......Page 161
12. Theorem of H.-E. Richert......Page 164
13. A conjecture on prime numbers......Page 166
14. Inequalities for the function π (x)......Page 170
15. The prime number theorem and its consequences......Page 175
1. Number of divisors......Page 179
2. Sums d(1)+d(2)+ ... +d(n)......Page 182
3. Numbers d(n) as coefficients of expansions......Page 186
4. Sum of divisors......Page 187
5. Perfect numbers......Page 195
6. Amicable numbers......Page 199
7. The sum σ'(1)+σ"(2)+ .., +σ"(n)......Page 201
8. The numbers σ"(n) as coefficients of various expansions......Page 203
9. Sums of summands depending on the natural divisors of a natural number n......Page 204
10. The Möbius function......Page 205
11. The Liouville function λ (n)......Page 209
1. Congruences and their simplest properties......Page 211
2. Roots of congruences. Complete set of residues......Page 216
3. Roots of polynomials and roots of congruences......Page 219
4. Congruences of the first degree......Page 222
5. Wilson's theorem and the simple theorem of Fermat......Page 224
6. Numeri idonei......Page 241
7. Pseudoprime and absolutely pseudoprime numbers......Page 242
8. Lagrange's theorem......Page 248
9. Congruences of the second degree......Page 252
1. Euler's totient function......Page 258
2. Properties of Euler's totient function......Page 270
3. The theorem of Euler......Page 273
4. Numbers which belong to a given exponent with respect to a given modulus......Page 276
5. Proof of the existence of infinitely many primes in the arithmetical progression nk+1......Page 281
6. Proof of the existence of the primitive root of a prime number......Page 285
7. An nth power residue for a prime modulus p......Page 289
8. Indices, their properties and applications......Page 292
1. Representation of natural numbers by decimals in a given scale......Page 298
2. Representations of numbers by decimals in negative scales......Page 303
3. Infinite fractions in a given scale......Page 304
4. Representations of rational numbers by decimals......Page 308
5. Normal numbers and absolutely normal numbers......Page 312
6. Decimals in the varying scale......Page 313
1. Continued fractions and their convergents......Page 317
2. Representation of irrational numbers by continued fractions......Page 319
3. Law of the best approximation......Page 325
4. Continued fractions of quadratic irrationals......Page 326
5. Application of the continued fraction for vD in solving the equations x2 – Dy2 = 1 and x2 – Dy2 = – 1......Page 342
6. Continued fractions other than simple continued fractions......Page 348
1. Legendre's symbol (D/P ) and its properties......Page 353
2. The quadratic reciprocity law......Page 359
3. Calculation of Legendre's symbol by its properties......Page 364
4. Jacobi's symbol and its properties......Page 365
5. Eisenstein's rule......Page 368
1. Some properties of Mersenne numbers......Page 373
2. Theorem of E. Lucas and D. H. Lehmer......Page 376
3. How the greatest of the known prime numbers have been found......Page 380
4. Prime divisors of Fermat numbers......Page 382
5. A necessary and sufficient condition for a Fermat number to be a prime......Page 388
1. Sums of two squares......Page 391
2. The average number of representations as sums of two squares......Page 394
3. Sums of two squares of natural numbers......Page 401
4. Sums of three squares......Page 404
5. Representation by four squares......Page 410
6. The sums of the squares of four natural numbers......Page 415
7. Sums of m ≥ 5 positive squares......Page 421
8. The difference of two squares......Page 423
9. Sums of two cubes......Page 425
10. The equation x3+y3 = z3......Page 428
11. Sums of three cubes......Page 432
12. Sums of four cubes......Page 435
13. Equal sums of different cubes......Page 437
14. Sums of biquadrates......Page 438
15. Waring's theorem......Page 440
1. Partitio numerorum......Page 444
2. Representations as sums of n non-negative summands......Page 446
3. Magic squares......Page 447
4. Schur's theorem and its corollaries......Page 452
5. Odd numbers which are not of the form 2k +p, where p is a prime......Page 458
1. Complex integers and their norm. Associated integers......Page 462
2. Euclidean algorithm and the greatest common divisor of complex integers......Page 466
3. The least common multiple of complex integers......Page 471
4. Complex primes......Page 472
5. The factorization of complex integers into complex prime factors......Page 476
6. The number of complex integers with a given norm......Page 478
7. Jacobi's four-square theorem......Page 482
Bibliography......Page 495
Author index......Page 518
Subject index......Page 524
Addendum and Corrigendum Inserted in July, 1987......Page 527
