The Chinese Remainder Theorem

Transcription

1 FORMALIZED MATHEMATICS Volume 6, Number 4, 1997 University of Białystok The Chinese Remainder Theorem Andrzej Kondracki AMS Management Systems Poland Warsaw Summary.ThearticleisatranslationofthefirstchaptersofabookWstęp do teorii liczb(eng. Introduction to Number Theory) by W. Sierpiński, WSiP, Biblioteczka Matematyczna, Warszawa, The first few pages of this book have already been formalized in MML. We prove the Chinese Remainder Theorem and Thue s Theorem as well as several useful number theory propositions. MML Identifier: WSIERP 1. The terminology and notation used in this paper are introduced in the following articles:[20],[16],[9],[14],[18],[1],[10],[13],[12],[15],[11],[17],[21],[6],[7], [2],[5],[3],[8],[4],and[19]. Forsimplicity,wefollowtherules: x, y, z, wdenoterealnumbers, a, b, c, d, e, f, gdenotenaturalnumbers, k, l, m, n, m 1, n 1 denoteintegers,and qdenotes a rational number. The following propositions are true: (1) If y 0,then ( x y )a = xa y a. (2) x 2 = x xand ( x) 2 = x 2. (3) ( x) 2 a = x 2 a and ( x) 2 a+1 = x 2 a+1. (4) If x 0,then x a Z = xa. (5) If x 0and y 0and d > 0and x d = y d,then x = y. (6) x >max(y, z)iff x > yand x > z. (7) If x 0and y z,then y x zand y z + x. (8) If x 0and y > zor x < 0and y z,then y > z + xand y x > z. Letusconsider a, b.thengcd(a,b)isanaturalnumber.letusobservethat the functor gcd(a, b) is commutative. Letusconsider m, n.then mgcdnisaninteger.letusobservethatthe functor m gcd n is commutative. 573 c 1997 University of Białystok ISSN

2 574 andrzej kondracki Letusconsider k, a.then k a isaninteger. Letusconsider a, b.then a b isanaturalnumber. We now state a number of propositions: (9) If k mand k n,then k m + n. (10) If k mand k n,then k m m 1 + n n 1. (11) If mgcdn = 1and kgcdn = 1,then m kgcdn = 1. (12) Ifgcd(a,b) = 1andgcd(c,b) = 1,thengcd(a c,b) = 1. (13) 0gcdm = m and 1gcdm = 1. (14) 1and karerelativeprime. (15) If kand larerelativeprime,then k a and larerelativeprime. (16) If kand larerelativeprime,then k a and l b arerelativeprime. (17) If kgcdl = 1,then kgcdl b = 1and k a gcdl b = 1. (18) m kiff m k. (19) If a b,then a c b c. (20) If a 1,then a = 1. (21) If d aandgcd(a,b) = 1,thengcd(d,b) = 1. (22) If k 0,then k liff l k isaninteger. (23) If a b c,then a band c b. Inthesequel f 1, f 2, f 3 arefinitesequences. Next we state two propositions: (24) If a Seglenf 2,then a Seglen(f 2 f 3 ). (25) If a Seglenf 3,thenlenf 2 + a Seglen(f 2 f 3 ). Let f 4 beafinitesequenceofelementsof Randletusconsider a.then f 4 (a) isarealnumber. Let f 5 beafinitesequenceofelementsof Zandletusconsider a.then f 5 (a) is an integer. Let f 6 beafinitesequenceofelementsof Nandletusconsider a.then f 6 (a) is a natural number. Let Dbeanonemptysetandlet D 1 beanonemptysubsetof D.Wesee thatthefinitesequenceofelementsof D 1 isafinitesequenceofelementsof D. Let Dbeanonemptyset,let D 1 beanonemptysubsetof D,andlet f 7, f 8 befinitesequencesofelementsof D 1.Then f 7 f 8 isafinitesequenceof elementsof D 1. Let Dbeanonemptysetandlet D 1 beanonemptysubsetof D.Then ε (D1 )isanemptyfinitesequenceofelementsof D 1. Zisanonemptysubsetof R. Forsimplicity,weadoptthefollowingconvention: D, D 1 arenonempty sets, v 1, v 2, v 3 aresets, f 6 isafinitesequenceofelementsof N, f 5, f 9 arefinite sequencesofelementsof Z,and f 4 isafinitesequenceofelementsof R.

3 the chinese remainder theorem 575 Letusconsider f 5.Then f 5 isaninteger.then f 5 isaninteger. Letusconsider f 6.Then f 6 isanaturalnumber.then f 6 isanatural number. Letusconsider a, f 1.Thefunctor f 1 ayieldingafinitesequenceisdefined as follows: (Def.1)(i) f 1 a = f 1 if a / domf 1, (ii) len(f 1 a) + 1 =len f 1 andforevery bholdsif b < a,then (f 1 a)(b) = f 1 (b)andif b a,then (f 1 a)(b) = f 1 (b + 1),otherwise. Letusconsider D,letusconsider a,andlet f 1 beafinitesequenceofelements of D.Then f 1 aisafinitesequenceofelementsof D. Letusconsider D,let D 1 beanonemptysubsetof D,letusconsider a,and let f 1 beafinitesequenceofelementsof D 1.Then f 1 aisafinitesequenceof elementsof D 1. One can prove the following propositions: (26) v 1 1 = εand v 1,v 2 1 = v 2 and v 1,v 2 2 = v 1 and v 1,v 2, v 3 1 = v 2,v 3 and v 1,v 2,v 3 2 = v 1,v 3 and v 1,v 2, v 3 3 = v 1,v 2. (27) If 1 aand a lenf 4,then (f 4 a) + f 4 (a) = f 4. (28) If a Seglen f 6 and f 6 (a) 0,then f6 f 6 (a) isanaturalnumber. (29) numqanddenqarerelativeprime. (30) If q 0and q = k aand a 0and kand aarerelativeprime,then k =numqand a =denq. (31) Ifthereexists qsuchthat a = q b,thenthereexists ksuchthat a = k b. (32) Ifthereexists qsuchthat a = q d,thenthereexists bsuchthat a = b d. (33) If e > 0and a e b e,then a b. (34) Thereexist m, nsuchthatgcd(a, b) = a m+b n. (35) Thereexist m 1, n 1 suchthat mgcdn = m m 1 + n n 1. (36) If m n kand mgcdn = 1,then m k. (37) Ifgcd(a,b) = 1and a b c,then a c. (38) If a 0and b 0,thenthereexist c, dsuchthatgcd(a,b) = a c b d. (39) If f > 0and g > 0andgcd(f,g) = 1and a f = b g,thenthereexists e suchthat a = e g and b = e f. Inthesequel x, y, z, tdenoteintegers. Next we state several propositions: (40) Thereexist x, ysuchthat m x+n y = kiff mgcdn k. (41) Suppose m 0and n 0and m m 1 + n n 1 = k.letgiven x, y.if n m x+n y = k,thenthereexists tsuchthat x = m 1 + t mgcd n and m y = n 1 t mgcd n. (42) Ifgcd(a, b) = 1and a b = c d,thenthereexist e, fsuchthat a = e d and b = f d.

4 576 andrzej kondracki (43) For every d such that for every a such that a Seglen f 6 holds gcd(f 6 (a),d) = 1holdsgcd( f 6,d) = 1. (44) Supposelenf 6 2andforall b, csuchthat b Seglenf 6 and c Seglenf 6 and b choldsgcd(f 6 (b),f 6 (c)) = 1.Letgiven f 5.Suppose lenf 5 =len f 6.Thenthereexists f 9 suchthatlenf 9 =len f 6 andforevery bsuchthat b Seglenf 6 holds f 6 (b) f 9 (b) + f 5 (b) = f 6 (1) f 9 (1) + f 5 (1). (45) If x < yand z wor x yand z > wor x < yand z > w,then x z < y w. (46) If a 0and agcdk = 1,thenthereexist b, esuchthat 0 band 0 e and b aand e aand a k b + eor a k b e. References [1] Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41 46, [2] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1): , [3] Grzegorz Bancerek and Piotr Rudnicki. Two programs for scm. Part I- preliminaries. Formalized Mathematics, 4(1):69 72, [4] JózefBiałasandYatsukaNakamura.DyadicnumbersandT 4topologicalspaces.Formalized Mathematics, 5(3): , [5] Czesław Byliński. Finite sequences and tuples of elements of a non-empty sets. Formalized Mathematics, 1(3): , [6] Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55 65, [7] Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1): , [8] Czesław Byliński. The sum and product of finite sequences of real numbers. Formalized Mathematics, 1(4): , [9] Krzysztof Hryniewiecki. Basic properties of real numbers. Formalized Mathematics, 1(1):35 40, [10] Andrzej Kondracki. Basic properties of rational numbers. Formalized Mathematics, 1(5): , [11] Jarosław Kotowicz. Real sequences and basic operations on them. Formalized Mathematics, 1(2): , [12] Rafał Kwiatek. Factorial and Newton coefficients. Formalized Mathematics, 1(5): , [13] Rafał Kwiatek and Grzegorz Zwara. The divisibility of integers and integer relative primes. Formalized Mathematics, 1(5): , [14] Jan Popiołek. Some properties of functions modul and signum. Formalized Mathematics, 1(2): , [15] Konrad Raczkowski. Integer and rational exponents. Formalized Mathematics, 2(1): , [16] Andrzej Trybulec. Tarski Grothendieck set theory. Formalized Mathematics, 1(1):9 11, [17] Andrzej Trybulec and Czesław Byliński. Some properties of real numbers. Formalized Mathematics, 1(3): , [18] Michał J. Trybulec. Integers. Formalized Mathematics, 1(3): , [19] Wojciech A. Trybulec. Pigeon hole principle. Formalized Mathematics, 1(3): , [20] Zinaida Trybulec and Halina Święczkowska. Boolean properties of sets. Formalized Mathematics, 1(1):17 23,

5 the chinese remainder theorem 577 [21] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1(1):73 83, Received December 30, 1997