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Introduction To Number Theory

Introduction To Number Theory. Seminar In Primality Testing. The Chineese Remainder theorem.

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Introduction To Number Theory

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  1. Introduction To Number Theory Seminar In Primality Testing

  2. The Chineese Remainder theorem

  3. 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 _________

  4. __ _ _____ ____ _ _____ _______ Consider 15=5*3 Now we look at18=6*3

  5. 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

  6. Proof: Uniqueness

  7. Proof: Finding a Solution (mod m*n)

  8. Lets check the solution: mod m: mod n:

  9. _______

  10. In general:

  11. Solution:

  12. Example:

  13. 1) Euler function There are 3 interesting cases:

  14. 2) Example: 0 1 2 3 4 5 6 7

  15. In general:

  16. 3)

  17. ____ Example:

  18. In general:

  19. Groups Definition: A Group is a set G with a binary operation on G, with the following properties:

  20. Some Examples:

  21. 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

  22. 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) !

  23. 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 )

  24. 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|

  25. An important remark: if H is a subgroup of a finite group G, then |H| |G|

  26. generated sub Groups (cont.)

  27. 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

  28. × • 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:

  29. Definition: let (G, ,e) be a group. The order of a G is: Lemma: Orders

  30. Orders (cont.)

  31. Proposition: Theorem (Euler): Fermat's Little Theorem:

  32. Definition: A Ring is a set R with 2 binary operations on R and 2 distinct elements 0 & 1, with the following properties: Rings

  33. The structure ={0,1,2,…m-1} with the binary operations Is a ring. Example

  34. Definition: A field is a set F with 2 binary operations on F and 2 distinct elements 0 & 1, with the following properties: Fields

  35. Proposition: m>1 is an integer. The following is equivalent: Fields(cont.)

  36. We will use 2 lemmas: Generators in finite fields: Theorem: If F is a finite filed, then F* is a cyclic group.

  37. In conclusion:

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