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CONTENTS

Preface................................ 5

CHAPTER I.
DIVISIBILITY AND INDETERMINATE EQUATIONS OF FIRST DEGREE

§ 1. Divisibility..................... 7
§ 2. Least common multiple..................... 10
§ 3. Greatest common divisor..................... 11
§ 4. Relatively prime numbers..................... 11
§ 5. Relation between the greatest common divisor and the least common multiple..................... 14
§ 6. Fundamental theorem of arithmetic..................... 14
§ 7. Proof of the formulae $(a_1, a_2,…, a_(n+1)) = ((a_1, a_2,…, a_n),a_(n+1))$ and $[a_1, a_2,…, a_(n+1)] = [[a_1, a_2,…,a_n],a_(n+1)]$..................... 18
§ 8. Rules for calculating the greatest common divisor of two numbers..................... 19
§ 9. Representation of rationals as simple continued fractions..................... 23
§ 10. Linear form of the greatest common divisor..................... 24
§ 11. Indeterminate equations of m variables and degree 1..................... 27
§ 12. Chinese Remainder Theorem..................... 31
§ 13. Thue Theorem..................... 33
§ 14. Square-free numbers..................... 33

CHAPTER II.
DIOPHANTINE ANALYSIS OF SECOND AND HIGHER DEGREES

§ 1. Diophantine equations of arbitrary degree and one unknown..................... 35
§ 2. Problems concerning Diophantine equations of two or more unknowns..................... 36
§ 3. The equation $x^2 + y^2 = z^2$..................... 38
§ 4. Integral solutions of the equation $x^2 + y^2 = z^2$ for each x-y = ± 1..................... 44
§ 5. Pythagorean triangles of the same area..................... 48
§ 6. On squares whose sum and difference are squares..................... 52
§ 7. The equation $x^4 + y^4 = z^2$..................... 58
§ 8. On three squares for which the sum of any two is a square..................... 61
§ 9. Congruent numbers..................... 63
§ 10. The equation $x^2 + y^2 + z^2 = t^2$..................... 67
§ 11. The equation xy = zt..................... 70
§ 12. The equation $x^4 - x^2y^2 + y^4 = z^2$..................... 73
§ 13. The equation $x^4+9x^2y^2 + 27y^4 = z^2..................... 75
§ 14. The equation $x^3 + y^3 = 2z^3$..................... 78
§ 15. The equation $x^3 + y^3 = az^3$ with a>2..................... 82
§ 16. Triangular numbers..................... 84
§ 17. The equation $x^2 - Dy^2 = 1$..................... 88
§ 18. The equations $x^2 + k = y^3$ where k is an integer..................... 99
§ 19. On some exponential equations and others..................... 106

CHAPTER III.
PRIME NUMBERS

§ 1. The primes. Factorization of a natural number m into primes..................... 110
§ 2. The Eratosthenes sieve. Tables of prime numbers..................... 114
§ 3. The differences between consecutive prime numbers..................... 115
§ 4. Goldbach's conjecture..................... 118
§ 5. Arithmetical progressions whose terms are prime numbers..................... 121
§ 6. Primes in a given arithmetical progression..................... 123
§ 7. Trinomial of Euler $x^2 + x + 41$..................... 125
§ 8. The conjecture H..................... 127
§ 9. The function π(x)..................... 130
§ 10. Proof of Bertrand's postulate (Theorem of Tchebycheff)..................... 131
§ 11. Theorem of H. F. Scherk..................... 140
§ 12. Theorem of H. E. Eichert..................... 143
§ 13. A conjecture on prime numbers..................... 145
§ 14. Inequalities for the function π(x)..................... 147
§ 15. The prime number theorem and its consequences..................... 152

CHAPTER IV.
NUMBER OF DIVISORS AND THEIR SUM

§ 1. Number of divisors..................... 156
§ 2. Sums d(1) + d(2) + … + d(n)..................... 159
§ 3. Numbers d(n) as coefficients of expansions..................... 163
§ 4. Sum of divisors..................... 164
§ 5. Perfect numbers..................... 171
§ 6. Amicable numbers..................... 175
§ 7. The sum σ(1) + σ(2) + … + σ(n)..................... 176
§ 8. The numbers σ(n) as coefficients of various expansions..................... 178
§ 9. Sums of summands depending on the natural divisors of a natural number n..................... 179
§ 10. Möbius function..................... 180
§ 11. Liouville function λ(n)..................... 184

CHAPTER V.
CONGRUENCES

§ 1. Congruences and their simplest properties..................... 186
§ 2. Roots of congruences. Complete set of residues..................... 191
§ 3. Roots of polynomials and roots of congruences..................... 194
§ 4. Congruences of the first degree..................... 196
§ 5. Wilson's theorem and the simple theorem of Fermat..................... 198
§ 6. Numeri idonei..................... 214
§ 7. Pseudoprime and absolutely pseudoprime numbers..................... 214
§ 8. Lagrange's theorem..................... 220
§ 9. Congruences of the second degree..................... 223

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

§ 1. Euler's totient function..................... 228
§ 2. Properties of Euler's totient function..................... 239
§ 3. The theorem of Euler..................... 241
§ 4. Numbers which belong to a given exponent with respect to a given modulu..................... 245
§ 5. Proof of the existence of infinitely many primes in the arithmetical progression nk+1..................... 248
§ 6. Proof of the existence of the primitive root of a prime number..................... 252
§ 7. An nth power residue for a prime modulus p..................... 256
§ 8. Indices, their properties and applications..................... 259

CHAPTER VII.
REPRESENTATION OF NUMBERS BY DECIMALS IN A GIVEN SCALE

§ 1. Representation of natural numbers by decimals in a given scale..................... 264
§ 2. Representations of numbers by decimals in negative scales..................... 269
§ 3. Infinite fractions in a given scale..................... 270
§ 4. Representations of rational numbers by decimals..................... 273
§ 5. Normal numbers and absolutely normal numbers..................... 277
§ 6. Decimals in the varying scale..................... 278

CHAPTER VIII.
CONTINUED FRACTIONS

§ 1. Continued fractions and their convergents..................... 282
§ 2. Representation of irrational numbers by continued fractions..................... 284
§ 3. Law of best approximation..................... 289
§ 4. Continued fractions of quadratic irrationals..................... 290
§ 5. Application of the continued fraction for √D in solving equations $x^2-Dy^2$ and $x^2-Dy^2=-1$..................... 305
§ 6. Continued fractions other than simple continued fractions..................... 310

CHAPTER IX.
LEGENDRE'S SYMBOL AND JACOBI'S SYMBOL

§ 1. Legendre's symbol (D/p) and its properties..................... 315
§ 2. The quadratic reciprocity law..................... 321
§ 3. Calculation of Legendre's symbol by its properties..................... 325
§ 4. Jacobi's symbol and its properties..................... 326
§ 5. Eisentein's rule..................... 329

CHAPTER X.
MERSENNE NUMBERS AND FERMAT NUMBERS

§ 1. Some properties of Mersenne numbers..................... 334
§ 2. Theorem of E. Lucas and D. H. Lehmer..................... 336
§ 3. How the greatest of the known prime numbers have been found..................... 340
§ 4. Prime divisors of Fermat numbers..................... 342
§ 5. A necessary and sufficient condition for a Fermat number to be a prime..................... 347
§ 6. How the fact that number $2^(2^{1945}) + 1$ is divisible by $5*2^{1947}+1$ was discovered..................... 349

CHAPTER XI.
REPRESENTATIONS OF NATURAL NUMBERS AS SUMS OF NON-NEGATIVE kth POWERS

§ 1. Sums of two squares..................... 351
§ 2. The average number of representations as sums of two squares..................... 354
§ 3. Sums of two squares of natural numbers..................... 360
§ 4. Sums of three squares..................... 363
§ 5. Representation by four squares..................... 368
§ 6. The sums of the squares of four natural numbers..................... 373
§ 7. Sums of m ≥ 5 positive squares..................... 378
§ 8. The difference of two squares..................... 380
§ 9. Sums of two cubes..................... 382
§ 10. The equation $x^3 + y^3 = z^3$..................... 384
§ 11. Sums of three cubes..................... 388
§ 12. Sums of four cubes..................... 391
§ 13. Equal sums of different cubes..................... 393
§ 14. Sums of biquadrates..................... 394
§ 15. Waring's theorem..................... 395

CHAPTER XII.
SOME PROBLEMS OF THE ADDITIVE THEORY OF NUMBERS

§ 1. Partitio numerorum..................... 400
§ 2. Representations as sums of n non-negative summands..................... 402
§ 3. Magic squares..................... 403
§ 4. Schur's theorem and its corollaries..................... 407
§ 5. Odd numbers which are not of the form $2^k+p$, where p is a prime..................... 412

CHAPTER XIII.
COMPLEX INTEGERS

§ 1. Complex integers and their norm. Associated integer..................... 416
§ 2. Euclidean algorithm and the greatest common divisor of complex integers..................... 420
§ 3. The least common multiply of complex integers..................... 424
§ 4. Complex primes..................... 425
§ 5. The factorization of complex integers into complex prime factors..................... 429
§ 6. The number of complex integers with a given norm..................... 431
§ 7. Jacobi's four-square theorem Bibliography Author index Subject index..................... 435

Bibliography..................... 488
Author index..................... 469
Subject index..................... 474