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Class 22: Primes

David Evans
November 15, 2017
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Class 22: Primes

cs2102: Discrete Mathematics
University of Virginia, Fall 2017

See course site for notes:
https://uvacs2102.github.io

David Evans

November 15, 2017
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  1. Plan This week: Number Theory and Cryptography Today: Primes Thursday:

    Asymmetric Cryptosystems Return Exam 2 at end of class today Remaining assignments: Problem Set 9 (out Sunday, due Dec 1) Problem Set Ѡ (posted today, (optionally) due Dec 4 Final Exam – 9am-noon, Thursday, Dec 7
  2. Recap from Class 20 Number theory: study of the integers

    Definition: divides ( | ) iff there is an integer such that = . Fundamental theorem of arithmetic: every positive number can be written uniquely as a product of primes: = ) ⋅ + ⋅ … ⋅ -where . ≤ .0) Introduced without proof – we will prove it today!
  3. [Number theorists] may be justified in rejoicing that there is

    one science at any rate, and that their own, whose very remoteness from ordinary human activities should keep it gentle and clean. (as quoted in MCS book)
  4. [Number theorists] may be justified in rejoicing that there is

    one science at any rate, and that their own, whose very remoteness from ordinary human activities should keep it gentle and clean. (as quoted in MCS book)
  5. [Number theorists] may be justified in rejoicing that there is

    one science at any rate, and that their own, whose very remoteness from ordinary human activities should keep it gentle and clean. (as written) But science works for evil as well as for good (and particularly, of course, in time of war); and both Gauss and less mathematicians may be justified in rejoicing that there is one science at any rate, and that their own, whose very remoteness from ordinary human activities should keep it gentle and clean.
  6. There is one comforting conclusions which is easy for a

    real mathematician. Real mathematics has no effects on war. No one has yet discovered any warlike purpose to be served by the theory of numbers or relativity, and it seems very unlikely that anyone will do so for many years.
  7. Note: is 1 prime? Fundamental theorem of arithmetic: every positive

    number can be written uniquely as a product of primes: = ) ⋅ + ⋅ … ⋅ -where . ≤ .0)
  8. Definition. A prime is a number greater than 1 that

    is divisible only by itself and 1. Fundamental theorem of arithmetic: every positive number can be written uniquely as a product of primes: = ) ⋅ + ⋅ … ⋅ -where . ≤ .0)
  9. Fundamental theorem of arithmetic: every positive number can be written

    uniquely as a product of primes: = ) ⋅ + ⋅ … ⋅ -where . ≤ .0) (1) There exists a factorization (Theorem 2.3.1). Every positive integer greater than one can be factored as a product of primes. (2) That factorization is unique.
  10. (1) There exists a factorization (Theorem 2.3.1). Every positive integer

    greater than one can be factored as a product of primes.
  11. Defining Rings A ring is a set, , with two

    binary operations that satisfy the three ring axioms: 1. R is Abelian (commutative) under the first operation (): ∀, , ∈ : associative: = commutative: = identity: ∃ ∈ : = inverse: ∃ ∈ : =
  12. Abelian Group is Abelian (commutative) under the first operation ():

    ∀, , ∈ : • associative: = • commutative: = • identity: ∃ ∈ : = • inverse: ∃ ∈ : = What is are examples of sets and operations that satisfy this property?
  13. Abelian Group is Abelian (commutative) under the first operation ():

    ∀, , ∈ : associative: = commutative: = identity: ∃ ∈ : = inverse: ∃ ∈ : = Which of these are Abelian groups: (1) = ℕ, = + (2) = ℕ, = × (3) = ℤ, = + (4) = ℚ, =× (5) = {, }, =
  14. 13 39 31 49 53 54 41 24 3 0

    10 20 30 40 50 <60 <70 <75 <80 <85 <90 <95 <100 >=100 Exam 2 Histogram
  15. Results by Question 0 (Name) 1 (revisit Exam 1) 2

    (State Machines) 3 (Recursive Definition) 4 (Structural Induction) 5 (Program Verification) 6 (Termination) 7 (Correctness) 8 (Infinite Cardinalities) Average 20 6.1 6.8 5.9 6.8 8.9 9.8 7.1 9.0 Median 20 7 7 6 8 9 10 7 10 For each problem, a good answer was worth 10 points.
  16. Results by Question 0 (Name) 1 (revisit Exam 1) 2

    (State Machines) 3 (Recursive Definition) 4 (Structural Induction) 5 (Program Verification) 6 (Termination) 7 (Correctness) 8 (Infinite Cardinalities) Average 20 6.1 6.8 5.9 6.8 8.9 9.8 7.1 9.0 Median 20 7 7 6 8 9 10 7 10 Note: we will ask a form of this question again on the final!
  17. Formula vs. Circuit NAND(NAND(, NAND(, )), NAND(, NAND(, )) 5

    NAND operations 4 NAND gates Connor Roos’ solution (exhaustive proof there are no 3-NAND solutions)
  18. Returning Exam 2 Group 4 Group 3 Group 2 Group

    1 qed4wg (Quinn) kai4vb (Kelsey) ea6dc (Elmo) aa8dp (Akanksha) rac4ad (Raina) kc4au (Kunal) eae4bf (Eliza) aap7sc (Alex) … … … … zpg9js (Zach) pw2hm (Peter) jyy3gx (Josh) dz7as (Daniel)