 Kolekcja Matematyczna   Prosimy o przesyłanie uwag na adres bwm@icm.edu.pl

Monografie Matematyczne

Tom 42

Elementary theory of numbers

Wacław Sierpiński

Warszawa 1964

Spis treści

 CHAPTER I DIVISIBILITY AND INDETERMINATE EQUATIONS OF FIRST DEGREE § 1. Divisibility § 2. Least common multiple § 3. Greatest common divisor § 4. Relatively prime numbers § 5. Relation between the greatest common divisor and the least common multiple § 6. Fundamental theorem of arithmetic § 7. Proof of the formulae ... § 8. Rules for calculating the greatest common divisor of two numbers § 9. Rrepresentation of rationals as simple continued fractions § 10. Linear form of the greatest common divisor § 11.FIndeterminate equations of m variables and degree 1 § 12. Chinese Remainder Theorem § 13. Thue Theorem § 14. Square-free numbers CHAPTER II DIOPHANTINE ANALYSIS OF SECOND AND HIGHER DEGREES § 1. Diophantine equations of arbitrary degree and one unknown § 2. Problems concerning Diophantine equations of two or more unknowns § 3. The equation x2 + y2 = z2 § 4 .Integral solutions of the equation ... § 5. Pythagorean triangles of the same area § 6. On squares whose sum and difference are squares § 7. The equation x4 + y4 = z2 § 8. On three squares for which the sum of any two is a square § 9. Congruent numbers § 10. The equation x2 + y2 + z2 = t2 § 11. The equation xy = zt § 12. The equation x4 - x2y2 + y4 = z2 § 13. The equation x4+9x2y2 + 27y4 = z2 § 14. The equation x3 + y3 = 2z3 § 15. The equation x3 + y3 = az3 with a>2 § 16. Triangular numbers § 17. The equation x2  Dy2 = 1 § 18. The equations x2 + k = y3 where k is an integer § 19. On some exponential equations and others CHAPTER III PRIME NUMBERS § 1. The primes. Factorization of a natural number m into primes § 2. The Eratosthenes sieve. Tables of prime numbers § 3. The differences between consecutive prime numbers § 4. Goldbach's conjecture § 5. Arithmetical progressions whose terms are prime numbers § 6. Primes in a given arithmetical progression § 7. Trinomial of Euler x2 + x + 41 § 8. The conjecture H § 9. The function ... § 10. Proof of Bertrand's postulate (Theorem of Tchebycheff) § 11. Theorem of H. F. Scherk § 12. Theorem of H. E. Eichert § 13. A conjecture on prime numbers § 14. Inequalities for the function ... § 15. The prime number theorem and its consequences CHAPTER IV NUMBER OF DIVISORS AND THEIR SUM § 1. Number of divisors § 2. Sums d(1) + d(2) + ... + d(n) § 3. Numbers d(n) as coefficients of expansions § 4. Sum of divisors § 5. Perfect numbers § 6. Amicable numbers § 7. The sum ... § 8. The numbers ... as coefficients of various expansions § 9. Sums of summands depending on the natural divisors of a natural number n § 10. Mobius function § 11. Liouville function ... CHAPTER V CONGRUENCES § 1. Congruences and their simplest properties § 2. Roots of congruences. Complete set of residues § 3. Eoots of polynomials and roots of congruences § 4. Congruences of the first degree § 5. Wilson's theorem and the simple theorem of Fermat § 6. Numeri idonei § 7. Pseudoprime and absolutely pseudoprime numbers § 8. Lagrange's theorem § 9. Congruences of the second degree CHAPTER VI EULER'S TOTIENT FUNCTION AND THE THEOREM OF EULER § 1. Euler's totient function § 2. Properties of Euler's totient function § 3. The theorem of Euler § 4. Numbers which belong to a given exponent with respect to a given modulus § 5. Proof of the existence of infinitely many primes in the arithmetical progression nk +1 § 6. Proof of the existence of the primitive root of a prime number § 7. An nth power residue for a prime modulus p § 8. Indices, their properties and applications CHAPTER VII REPRESENTATION OF NUMBERS BY DECIMALS IN A GIVEN SCALE § 1.Representation of natural numbers by decimals in a given scale § 2. Representations of numbers by decimals in negative scales § 3. Infinite fractions in a given scale § 4. Representations of rational numbers by decimals § 5. Normal numbers and absolutely normal numbers § 6. Decimals in the varying scale CHAPTER VIII CONTINUED FRACTIONS § 1. Continued fractions and their convergents § 2. Representation of irrational numbers by continued fractions § 3. Law of best approximation § 4. Continued fractions of quadratic irrationals § 5. Application of the continued fraction for ... § 6. Continued fractions other than simple continued fractions CHAPTER IX LEGENDRE'S SYMBOL AND JACOBI'S SYMBOL § 1. Legendre's symbol (D-p) and its properties § 2. The quadratic reciprocity law § 3. Calculation of Legendre's symbol by its properties § 4. Jacobi's symbol and its properties § 5. Eisentein's rule CHAPTER X MERSENNE NUMBERS AND FERMAT NUMBERS § 1. Some properties of Mersenne numbers § 2. Theorem of E. Lucas and D. H. Lehmer § 3. How the greatest of the known prime numbers have been found § 4. Prime divisors of Fermat numbers § 5. A necessary and sufficient condition for a Fermat number to be a prime § 6. How the fact that number ... CHAPTER XI REPRESENTATIONS OF NATURAL NUMBERS AS SUMS OF NON-NEGATIVE kth POWERS § 1. Sums of two squares § 2. The average number of representations as sums of two squares § 3. Sums of two squares of natural numbers § 4. Sums of three squares § 5. Representation by four squares § 6. The sums of the squares of four natural numbers § 7. Sums of m > 5 positive squares § 8. The difference of two squares § 9. Sums of two cubes § 10. The equation x3 + y3 = z3 § 11. Sums of three cubes § 12. Sums of four cubes § 13. Equal sums of different cubes § 14. Sums of biquadrates § 15. Waring's theorem CHAPTER XII SOME PROBLEMS OF THE ADDITIVE THEORY OF NUMBERS § 1. Partitio numerorum § 2. Representations as sums of n non-negative summands § 3. Magic squares § 4. Schur's theorem and its corollaries § 5. Odd numbers which are not of the form 2k+p, where p is a prime CHAPTER XIII COMPLEX INTEGERS § 1. Complex integers and their norm. Associated integer § 2. Euclidean algorithm and the greatest common divisor of complex integers § 3. The least common multiply of complex integers § 4. Complex primes § 5. The factorization of complex integers into complex prime factors § 6. The number of complex integers with a given norm § 7. Jacobi's four-square theorem Materiały redakcyjne Preface, bibliography, author index, subject index, contents, coreigendum