Class 22: Primes cs2102: Discrete Mathematics | F17 uvacs2102.github.io David Evans University of Virginia

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

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!

[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)

G. H. Hardy A Mathematician’s Apology 1940

The Man who Knew Infinity 2015 Movie Srinivasa Ramanujan (Dev Patel) G. H. Hardy (Jeremy Irons)

[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)

[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.

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.

G. H. Hardy A Mathematician’s Apology 1940

Theorem: There are infinitely many prime numbers.

Theorem: There are infinitely many prime numbers.

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Note: is 1 prime? Fundamental theorem of arithmetic: every positive number can be written uniquely as a product of primes: = ) ⋅ + ⋅ … ⋅ -where . ≤ .0)

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)

Visualizing Integers

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.

(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.

Groups, Rings, and Fields

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: ∃ ∈ : =

Niels Henrik Abel 1802-1829

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?

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) = {, }, =

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Exam 2

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

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.

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!

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Class 5 Challenge 5

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)

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)