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Differential inequalities

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Szarski J.

Instytut Matematyczny Polskiej Akademii Nauk

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

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1 Szarski J.

1 1965

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Differential inequalities