Math 412: Number Theory Lecture 6: congruence system and

Transcription

1 Math 412: Number Theory Lecture 6: congruence system and classes Gexin Yu College of William and Mary

2 Chinese Remainder Theorem Chinese Remainder Theorem: let m 1, m 2,..., m k be pairwise coprimes. Then for any integers a 1, a 2,..., a k, the system of linear equations x a 1 (mod m 1 ) x a 2 (mod m 2 )... x a k (mod m k ) has a solution. Furthermore, the solution is x M 1 M 1 1 a M k M 1 k a k (mod m), where m = m 1 m 2... m k = m j M j and M 1 j M j 1 (mod m j ).

3 Solving polynomial congruences Consider f (x) = a n x n + + a 1 x + a 0 0 (mod m) with a i Z. We may consider the system of congruence equations f (x) 0 (mod p r ), where p r is a prime power of m, then use Chinese Remainder Theorem. Hensel s Theorem: If c is a solution to f (x) 0 (mod p r 1 ), then the solution to f (x) 0 (mod p r ) with x c (mod p r 1 ) are x c + y i p r 1 (mod p r ), where y y 1,..., y l (mod p) are solutions to f (c)y f (c) p r 1 (mod p)

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5 Congruence systems Ex: Find the solutions to the following congruences: 3x + 4y 5 (mod 13) 2x + 5y 7 (mod 13)

6 Thm: Let a, b, c, d, e, f and m be integers with m > 0, and (, m) = 1, where = ad bc. Then the system of congruences ax + by e (mod m) cx + dy f (mod m) has a unique solution modulo m, given by x 1 (de bf ) (mod m) y 1 (af ce) (mod m)

7 congruent matrices Def: Let A = (a ij ), B = (b ij ) be n k matrices with integer entries. Then A is congruent to B modulo m if a ij b ij (mod m) for all 1 i n and 1 j k.

8 congruent matrices Def: Let A = (a ij ), B = (b ij ) be n k matrices with integer entries. Then A is congruent to B modulo m if a ij b ij (mod m) for all 1 i n and 1 j k. Thm: If A, B are n k matrices with A B (mod m), C is a k p matrix, and D is a p n matrix, all with integer entries, then AC BC (mod m) and DA DB (mod m)

9 Inverse of a matrix If A and B are n n matrices of integers and AB BA I (mod m), where I is the identity matrix. Then B is said to be an inverse of A modulo m.

10 Inverse of a matrix If A and B are n n matrices of integers and AB BA I (mod m), where I is the identity matrix. Then B is said to be an inverse of A modulo m. The inverse of an n n matrix, if exists, is unique.

11 Inverse of a matrix If A and B are n n matrices of integers and AB BA I (mod m), where I is the identity matrix. Then B is said to be an inverse of A modulo m. The inverse of an ( n ) n matrix, if exists, is unique. a b Thm: Let A = be a matrix with integers such that c d = det(a) = ad bc is a coprime to m. Then ( ) A 1 = 1 d b c a

12 Ex: Find the inverse of A = ( )

13 Congruent classes A complete system of residues modulo m is a set of integers such that every integer is congruent modulo m to exactly one integer of the set.

14 Congruent classes A complete system of residues modulo m is a set of integers such that every integer is congruent modulo m to exactly one integer of the set. Ex: A set of m incongruent integers modulo m forms a complete set of residues modulo m.

15 Congruent classes A complete system of residues modulo m is a set of integers such that every integer is congruent modulo m to exactly one integer of the set. Ex: A set of m incongruent integers modulo m forms a complete set of residues modulo m. Ex: If r 1,..., r m is a complete system of residues modulo m, and if a N and (a, m) = 1, then ar 1 + b, ar 2 + b,..., ar m + b is a complete system of residues modulo m for any integer b.

16 Ex: let n 1, and b has no prime divisor less than or equal to n. Then for any a N, n! a(a + b)(a + 2b)... (a + (n 1)b)