Elementary Theory of Numbers : Second English Edition (edited by A. Schinzel)
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 G
eBook, English, 2014
Elsevier Science, Amsterdam, 2014
1 online resource (527 pages)
9780080960197, 0080960197
1058522299
Front Cover; Elementary Theory of Numbers; Copyright Page; Contents; Author's Preface; Editor's Preface; 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 (al, a2 ..., an+1) = (al, a2 ..., an), an+1) and [a1, a2 ..., an+1] = [[a1, a2 ..., an], an+1]; 8. Rules for calculating the greatest common divisor of two numbers. 9. Representation of rationals as simple continued fractions10. Linear form of the greatest common divisor; 11. Indeterminate equations of m variables and degree 1; 12. Chinese Remainder Theorem; 13. Thue's Theorem; 14. Square-free numbers; Chapter II Diophantlne 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 x2+y2 = z2 for which x-y = 1; 5. Pythagorean triangles of the same area. 6. On squares whose sum and difference are squares7. 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 primes2. 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 s (x); 10. Proof of Bertrand's Postulate (Theorem of Tchebycheff); 11. Theorem of H.F. Scherk; 12. Theorem of H.-E. Richert; 13. A conjecture on prime numbers; 14. Inequalities for the function s (x). 15. The prime number theorem and its consequencesChapter 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 v'(1)+v""(2)+ .., +v""(n); 8. The numbers v""(n) as coefficients of various expansions; 9. Sums of summands depending on the natural divisors of a natural number n; 10. The Möbius function; 11. The Liouville function n (n); Chapter V Congruences; 1. Congruences and their simplest properties
2. Roots of congruences. Complete set of residues