Elements of Number Theory

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elem ents of number theory translated from the fifth revised edition by saul kravetz

ELEMENTS OF NUMBER THEORY

ELEMENTS OF NUMBER THEORY

BY I. M. VINOGRADOV

T ra n s la te d from the Fifth R evised Edition by Saul Kravetz

DOVER PUBLICATIONS, INC.

C o pyright 1954 by D over P u b l i c a t i o n s , Inc.

F i r s t E n g li s h t r a n s l a t i o n of the F if th R u s s i a n ed itio n o f 1949.

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M an u factu red in th e U n ited S t a t e s of Am

CONTENTS P reface

vii C hapter I

DIVISIBILITY THEORY § 1. B asic C oncepts and Theorem s ( 1). § 2 . The G reatest Common D ivisor (2). § 3 . The L e a s t Common Multiple (7). § 4 . The Relation of E u c lid ’s Algorithm to Continued F ractions (8). § 5 . Prime Numbers (14). § 6 . T he U nicity of Prime De­ composition (15). Problem s for C hapter I (17). Numerical E x e r c is e s for Chapter I (20). C hapter II

IMPORTANT NUMBER-THEORETICAL FUNCTIONS §1. The F u n ctio n s { x}»x (21). § 2. Su ms Extended over the Divisors of a Number (.22). §3. The Mobius Function (24). §4. The E uler Function (26). Problems for C hapter II (28). Numerical E x e r c is e s for Chapter II (40). Chapter III .

CONGRUENCES § 1. B asic C oncepts (41). § 2. P ro p erties of Congruences Similar to those of Equation® (42). §3. Further P ro p erties of Congruences (44). §4. Complete Systems of R esid u e s (45). §5. Reduced S ystem s of R esid u e s (47). § 6. The Theorem s of ill

Euler and Ferm at (48). Problem s for Chapter III (49). Numeri­ cal E x e r c is e s for Chapter III (58). C hapter IV

CONGRUENCES IN ONE UNKNOWN § 1. B a s ic C oncepts (59). § 2. C ongruences of the F irs t Degree (60). §3. System s of Congruences of the F ir s t Degree (63). §4. C ongruences of Arbitrary Degree with Prime Modu­ lus (65). § 5. Cong ruences of Arbitrary Degree with Com­ posite Modulus (66). Problem s for C hapter IV (71). Numerical E x e r c is e s for C hapter IV (77). C hapter V

CONGRUENCES OF SECOND DEGREE §1. General T heorem s (79). § 2. The Legendre Symbol (81). §3. The Jacobi Symbol (87). § 4 . The C a s e of Composite Moduli (91). Problem s for C hapter V (95). Numerical Exer­ c i s e s for Chapter V (103). C hapter VI

PRIMITIVE ROOTS AND INDICES § 1. General Theorem s (105). §2. Prim itive Roots Modulo p a and 2pa (106). §3. E valuation of Prim itive Roots for the Moduli p a and 2pa (108). § 4. Indices for the Moduli p a and 2pa (110). § 5 . C on seq u en ce s of the P receding Theory (113). § 6 . Indices Modulo 2a (116). §7. Indices for Arbitrary Com­ posite Modulus (119). Problem s for Chapter VI (121). Numeri­ cal E x e r c is e s for Chapter VI ( 130).

SOLUTIONS OF THE PROBLEMS Solutions for Chapter I Solutions for C hapter III Solutions for Chapter V IV

(133). Solutions for Chapter II (139). (161). Solutions for Chapter IV (178). (187). Solutions for Chapter VI (202).

ANSWERS TO THE NUMERICAL EXERCISES Answers for Chapter I A nsw ers for C hapter III A nsw ers for C hapter V

(217). (218). (218).

A nsw ers for C hapter II (217). A nsw ers for C hapter IV (218). Answers for C hapter VI (219).

TABLES O F IN D IC E S .....................................................................

220

TA BLES OF PRIMES