1. 1 CMSC 250 Chapter 3, Number Theory

2. 2 CMSC 250 Introductory number theory l A good proof should have: –a statement of what is to be proven –"Proof:" to indicate where the proof starts –a clear indication of flow –a clear indication of the reason for each step –careful notation, completeness and order –a clear indication of the conclusion

3. 3 CMSC 250 Some definitions l Z is the integers l Q is the rational numbers (quotients of integers) r  Q  (  a,b  Z) [(r = a / b)  (b  0)] l Irrational numbers are those which are not rational l R is the real numbers l A superscript of + indicates the positive portion only of one of these sets of numbers l A superscript of – indicates the negative portion only of one of these sets of numbers l Other superscripts can be used, such as Z even, Z odd, Q >5 l We can define the closure of these sets for an operation Z is closed under what operations?

4. 4 CMSC 250 Integer definitions l An even integer n  Z even  (  k  Z) [n = 2k] l An odd integer n  Z odd  (  k  Z) [n = 2k + 1] l A prime integer (Z >1 ) n  Z prime  (  r,s  Z + ) [ (n = r  s)  (r = 1)  (s = 1)] l A composite integer (Z >1 ) n  Z composite  (  r,s  Z + ) [(n = r  s)  (r  1)  (s  1)]

5. 5 CMSC 250 Constructive proof of existence l How can we prove the following?

6. 6 CMSC 250 Discrete Structures CMSC 250 Lecture 13 February 25, 2008

7. 7 CMSC 250 Methods of proving universally quantified statements l Method of exhaustion –prove the statement is true for each and every member of the domain –(  r  Z + )[23 < r < 29  (  p,q  Z + ) [(r = p  q)  (p  q)]] l Generalizing from the generic particular –suppose that x is a particular but arbitrarily-chosen element of the domain –show that x satisfies the property –then the result holds for every element of the domain –example: (  r  Z) [r  Z even  r 2  Z even ]

8. 8 CMSC 250 Examples of generalizing from the generic particular l The product of any two odd integers is also odd. –(  m,n  Z) [(m  Z odd  n  Z odd )  m  n  Z odd ] l The product of any two rationals is also rational. –(  m,n  Q) [m  n  Q]

9. 9 CMSC 250 Discrete Structures CMSC 250 Lecture 14 February 27, 2008

10. 10 CMSC 250 Disproof by counterexample l (  r  Z) [r 2  Z +  r  Z + ] –counterexample: r 2 = 9  r = –3 r 2  Z + since 9  Z + so the antecedent is true but r  Z + since –3  Z + so the consequent is false this means the implication is false for r = –3, so this is a valid counterexample l When you give a counterexample you must justify that it is a valid counterexample, by showing the algebra (or other interpretation needed) to support your claim

11. 11 CMSC 250 Divisibility l Definition: d | n  (  k  Z)[n = d × k] –n is divisible by d –n is a multiple of d –d is a divisor of n –d divides n l Results involving divisibility: (  a, b  Z >1 )[a | b → a | (b + 1)] (  x  Z >1 )(  p  Z prime )[p | x] –note another proof method would be used here, proof by division into cases

12. 12 CMSC 250 Proof by contrapositive For all positive integers n, if n does not divide a number of which d is a factor, then n can not divide d. (  n,d,c  Z + ) [n | dc  n | d] (  n,d,c  Z + ) [n | d  n | dc]

13. 13 CMSC 250 Discrete Structures CMSC 250 Lecture 15 February 29, 2008

14. 14 CMSC 250 Proof by contradiction l The number of primes is infinite l Assume this is false, that there is some largest prime number, so there are only n different prime numbers l Consider the number

15. 15 CMSC 250 Prime factored form l The Unique Factorization Theorem (Theorem 3.3.3) Given any integer n > 1, (  k  Z)(  p 1,p 2,…p k  Z prime )(  e 1,e 2,…e k  Z + )[n = p 1 e1 × p 2 e2 × p 3 e3 × …× p k ek ] where the p’s are distinct and any other expression of n is identical to this except maybe in the order of the factors. l Standard factored form n = p 1 e1 × p 2 e2 × p 3 e3 × … × p k ek p i < p i+1 l (  m  Z)[8×7×6×5×4×3×2×m = 17×16×15×14×13×12×11×10] –does 17 | m ??

16. 16 CMSC 250 More integer definitions l div and mod operators n div d- integer quotient for n mod d- integer remainder for (n div d = q) ^ (n mod d = r)  n = d × q + r where n  Z  0, d  Z +, r  Z, q  Z, 0  r < d l Relating “mod” to “divides”: d | n  0 = n mod d  0  d n l Definition of equivalence in a mod: x  d y  d | (x–y) (note: their remainders are equal) sometimes written as x  y mod d, meaning (x  y) mod d

17. 17 CMSC 250 Discrete Structures CMSC 250 Lecture 16 March 3, 2008

18. 18 CMSC 250 Modular arithmetic (Theorem 10.4.3) l Let a, b, c, d, n  Z, and n > 1. Suppose a  c (mod n) and b  d (mod n). Then: 1.(a + b)  (c + d) (mod n) 2.(a – b)  (c – d) (mod n) 3.ab  cd (mod n) 4.a m  c m (mod n) for all integers m l Proof using this definition: (  m  Z + )(  a,b  Z)[a  m b  (  k  Z) [a = b + km]]

19. 19 CMSC 250 Discrete Structures CMSC 250 Lecture 18 March 7, 2008

20. 20 CMSC 250 Floor and ceiling l Definitions: –n is the floor of x where x  R  n  Z  x  = n  n  x < n+1 –n is the ceiling of x where x  R  n  Z  x  = n  n–1 < x  n l Proofs using floor and ceiling: (  x,y  R) [  x+y  =  x  +  y  ] (  x  R)(  y  Z)[  x+y  =  x  + y ]

21. 21 CMSC 250 More proof by division into cases The floor of (n/2) is either a) n/2 when n is even or b) (n–1)/2 when n is odd

22. 22 CMSC 250 Discrete Structures CMSC 250 Lecture 19 March 10, 2008

23. 23 CMSC 250 The quotient remainder theorem (  n  Z)(  d  Z + )(  q,r  Z)[(n = dq + r)  (0  r < d)] Proving definition of equivalence in a mod using the quotient remainder theorem This means prove that if [m  d n], then [d | (n-m)] where m,n  Z and d  Z +

24. 24 CMSC 250 Proof by division into cases again (  n  Z) [3 | n  n 2  3 1] or, alternatively, (  n  Z) [3 | n  n 2  1 (mod 3)]

25. 25 CMSC 250 Steps toward proving the unique factorization theorem l Every integer greater than or equal to 2 has at least one prime that divides it l For all integers greater than 1, if a | b, then a | (b+1) l There are an infinite number of primes

26. 26 CMSC 250 Discrete Structures CMSC 250 Lecture 20 March 12, 2008

27. 27 CMSC 250 Using the unique factorization theorem l Prove that (  a  Z + )(  q  Z prime ) [q | a 2  q | a] l Prove that

28. 28 CMSC 250 Discrete Structures CMSC 250 Lecture 21 March 14, 2008

29. 29 CMSC 250 Summary of proof methods l Constructive proof of existence l Proof by exhaustion l Proof by generalizing from the generic particular l Proof by contraposition l Proof by contradiction l Proof by division into cases

30. 30 CMSC 250 Errors in proofs l Arguing from example for universal proof l Misuse of variables l Jumping to the conclusion (missing steps) l Begging the question l Using "if" about something that is known