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Introduction To Number Theory. Seminar In Primality Testing. The Chineese Remainder theorem.
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Introduction To Number Theory Seminar In Primality Testing
The Chineese Remainder theorem
The original form of the theorem , was contained in a third-century AD book(孙子算经 The Mathematical Classic by Sun Zi)by Chinese mathematician Sun Tzu and later republished in a 1247 book by Qin Jiushao _________
__ _ _____ ____ _ _____ _______ Consider 15=5*3 Now we look at18=6*3
Theorem : Suppose m, n are co-prime integers. So the next system of simultaneous congruences has a solution. Proof Furthermore, the solution is unique modulo n*m
Proof: Finding a Solution (mod m*n)
Lets check the solution: mod m: mod n:
1) Euler function There are 3 interesting cases:
2) Example: 0 1 2 3 4 5 6 7
____ Example:
Groups Definition: A Group is a set G with a binary operation on G, with the following properties:
A more concrete example: • 1 2 4 5 7 8 • 1 2 4 5 7 8 • 2 4 8 1 5 7 • 4 8 7 2 1 5 • 5 5 1 2 7 8 4 • 7 7 5 1 8 4 2 • 8 8 7 5 4 2 1
Definition: A Group (G, ,e) is Commutative (or Abelian) if ab= b a for all a, b G Commutative Groups In the previous examples, (a), (b), (c) & (d) Are commutative, but example (e) isn’t a commutative group (since matrix multiplication isn’t a commutative operation) !
Some simple Observations: (a) In a group, there is exactly one neutral element (will be called 1 from now) (b) In a group, there is exactly one inverse element for every a G (denoted as ) (c )
Sub Groups: Definition: Subgroup ! If is a finite group, and H is a subset of G, then we only need to satisfy conditions (i) & (ii) An important remark: if H is a subgroup of a finite group G, then |H| |G|
An important remark: if H is a subgroup of a finite group G, then |H| |G|
generated sub Groups (cont.)
Cyclic Groups Definition: A Group (G, ,e) is Cyclic if there exists an a G, s.t G=< a >. This a is called Generating element of G
× • 1 5 7 8 4 2 • 1 5 7 8 4 2 • 5 5 7 8 4 2 1 • 7 7 8 4 2 1 5 • 8 8 4 2 1 5 7 • 4 4 2 1 5 7 8 • 2 2 1 5 7 8 4 9 Examples:
Definition: let (G, ,e) be a group. The order of a G is: Lemma: Orders
Proposition: Theorem (Euler): Fermat's Little Theorem:
Definition: A Ring is a set R with 2 binary operations on R and 2 distinct elements 0 & 1, with the following properties: Rings
The structure ={0,1,2,…m-1} with the binary operations Is a ring. Example
Definition: A field is a set F with 2 binary operations on F and 2 distinct elements 0 & 1, with the following properties: Fields
Proposition: m>1 is an integer. The following is equivalent: Fields(cont.)
We will use 2 lemmas: Generators in finite fields: Theorem: If F is a finite filed, then F* is a cyclic group.