class 20

number theory

Transcript

1. cs2102: Discrete Mathematics Class 20: Number Theory David Evans, Mohammad Mahmoody University of Virginia
2. Plan + Reminders Today: number theory Deadline: PS8 this Friday 6:29pm. Tuesday: Review before the exam Next Thursday: 2nd Exam.
3. Clarification
4. Number Theory • A study of integer numbers ℤ = … , −2, −1, 0, 1, 2, … and their properties. All numbers in today’s class integers unless said otherwise.. • Main operations: 1. Addition 2. Multiplication 3. Division..?..
5. Divisibility • If there is such that = then:
6. More Examples • ∣ , ∣ → ? • ∣ , ∣ →? ∣ + for all , ?
7. Primes • > 1 is prime, if its (positive) divisors are 1, • Other > 1 are called composite.
8. Fundamental Theorem of Arithmetic • Every positive > 1 can be uniquely written as: = 1 ⋅ 2 ⋅ … where 1. ′ are all prime 2. ≤ +1 • or equivalently: = 1 1 ⋅ 2 2 ⋅ … where 1. ′ are all prime 2. < +1 3. ’s are all positive integers
9. Which one of these is true? 1. Every even number > 2 is sum of + for primes , . 2. There are infinitely many primes where + 2 is also prime. 3. For > 2 there are no integers , , where + =
10. 1. Every even number > 2 is sum of + for primes , . Goldbach conjecture. Verified up to ~1019 2. There are infinitely many primes where + 2 is also prime. Twin primes conjecture. Proved: infinite pairs , + 2 ,both *products* of two primes 3. For > 2 there are no integers , , where + = Fermat’s last theorem..
11. Fermat’s last theorem Pierre de Fermat 1607-1665 French Lawyer (and mathematician) Conjectured by Fermat in 1637 in the margin of Arithmetica where he says the proof is too large to fit in the margin
12. Fermat’s last theorem Conjectured by Fermat in 1637 in the margin of Arithmetica where he says the proof is too large to fit in the margin Proved by Andrew Wiles (then Princeton, now Oxford) in 1994 He was 41 when proved this, so could not get Fields medal for that..
13. Euclid’s algorithm
14. Greatest Common Divisor • For , > 1, = gcd(, ) if > 1 is the greatest common divisor ∣ , ∣ • Example: what is gcd(, ) if ∣ ? • Example: What is gcd(, ) if = and = for primes , ,
15. How to find gcd , in general?
16. Division Theorem • For all , there are , where = ⋅ + and 0 ≤ < || • = (, ) could be positive or negative • = (, ) is always non-negative • Example:
17. Useful Lemma • gcd , = gcd(, , ) • Why true? • Why useful?
18. Euclid’s gcd algorithm Assuming > > 0 • Def Euclid_gcd (m, n) while > 0 = , = = return
19. What is the underlying State-Machine? • Def Euclid_gcd (m, n) while > 0 = , = = return
20. What is a good preserved invariant? • Def Euclid_gcd (m, n) while > 0 = , = = return
21. Why does it terminate? • Def Euclid_gcd (m, n) while > 0 = , = = return
22. Charge • PS8 Due Friday (6:29pm) • Next week: – Tuesday: Review for Exam – Thursday: 2nd Exam