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Chapter 3: Elementary Number Theory and Methods of Proofs

Chapter 3: Elementary Number Theory and Methods of Proofs. 3.1-.3.4 Direct Methods and Counterexamples Introduction Rational Numbers Divisibility Division Algorithm. Instructor: Hayk Melikya melikyan@nccu.edu. Basic Definitions.

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Chapter 3: Elementary Number Theory and Methods of Proofs

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  1. Chapter 3: Elementary Number Theory and Methods of Proofs • 3.1-.3.4 Direct Methods and Counterexamples • Introduction • Rational Numbers • Divisibility • Division Algorithm Instructor: Hayk Melikya melikyan@nccu.edu

  2. Basic Definitions Definition: An integer n is an even number if there exists an integer k such that n = 2k. Symbolically: Let Even(n) := “an integer n is even”: E(n) = (k  Z)( n = 2k) . Def: An integer n is an odd number if there exists an integer k such that n = 2k+1. Symbolically: Let O(n) := “an integer is odd”: Odd(n) = (k  Z)( n = 2k +1) . Def: An integer n is a prime number if and only if n>1 and if n=rs for some positive integers r and s then r=1 or s=1.

  3. Primes and Composites Def: An integer n is a prime number if and only if n>1 and if n=rs for some positive integers r and s then r=1 or s=1. Symbolically: Prime(n):= n is prime  positive integers r and s, if n = rs then r =1 or s =1 Def: A positive integer n is a composite if and only if n=rs for some positive integers r and s then r ≠ 1 and s ≠ 1. • Symbolically: Cpmposite(n):= n is compoeite  positive integers r and s, • such that n = rs and r ≠ 1 and s ≠ 1 • The RSA Challenge (up to US$200,000) • http://www.rsasecurity.com/rsalabs/challenges/factoring/numbers.html Examples: Find the truth value of the following prpopositions E(6), P(12), C(17)

  4. Existential Statements x P(x) Proofs: • Constructive • Construct an example of such a such that P(a) is true • Non-constructive • By contradiction • Show that if such x does NOT exist than a contradiction can be derived

  5. Example Let G(n):= a  b ((a+b=n)  Prime(a)  Prime(b)) Prove that (nN)G(n) Proof: • n=210 • a=113 • b=97 Piece of cake… What about(nN) G(n) ( many Million $ baby)

  6. Universal Statements • x P(x) • x [Q(x)  R(x)] • Proof techniques: • Exhaustion • By contradiction • Assume the statement is not true • Arrive at a contradiction • Direct • Generalizing from an arbitrary particular member

  7. ├ (xU)P(x) To prove a theorem of the form (xU)P(x) (same as ├ (xU)P(x)) which states “for all elements x in a given universe U, the proposition P(x) is true” we select an arbitrary aUfrom the universe, and then prove the assertion P(a). Then by Universal generalization we conclude P(a)├ (xU)P(x) For arguments of the form├ x [Q(x)  R(x)]

  8. Example 1 Exhaustion: • Any even number between 4 and 30 can be written as a sum of two primes: • 4=2+2 • 6=3+3 • 8=3+5 • … • 30=11+19 Works for finite domains only What if I want to prove that for any integer n the product of n and n+1 is even? Can I exhaust all integer values of n?

  9. Example 2 Theorem: (nZ)( even(n*(n+1)) ) Proof: • Consider a particular but arbitrary chosen integer n • n is odd or even • Case 1: n is odd • Then n=2k+1, n+1=2k+2 • n(n+1) = (2k+1)(2k+2) = 2(2k+1)(k+1) = 2p for some integer p • So n(n+1) is even • Case 2: n is even • Then n=2k, n+1=2k+1 • n(n+1) = 2k(2k+1) = 2p for some integer p • So n(n+1) is even

  10. Fallacy Generalizing from a particular but NOT arbitrarily chosen example I.e., using some additional properties of n Example: • “all odd numbers are prime” • “Proof”: • Consider odd number 3 • It is prime • Thus for any odd nprime(n) holds Such “proofs” can be given for correct statements as well!

  11. Prevention • Try to stay away from specific instances (e.g., 3) • Make sure that you are not using any additional properties of n considered • Challenge your proof • Try to play the devil’s advocate and find holes in it… • Using the same letter to mean different things • Jumping to a conclusion • Insufficient justification • Begging the question assuming the claim first

  12. Rational Numbers A real number is rational iff it can be represented as a ratio/quotient/fraction of integers a and b(b0) • rR [rQ a,bZ [r=a/b & b0]] Notes: • a is numerator • b is denominator • Any rational number can be represented in infinitely many ways • The fractional part of any rational number written in any natural radix has a period in it

  13. Rational or not? • -12 • -12/1 • 3.1459 • 3+1459/10000 • 0.56895689568956895689… • 5689/9999 • 1+1/2+1/4+1/8+… • 2 • 0 • 0/1

  14. Theorem 1 Any number with a periodic fractional part in a natural radix representation is rational Proof: • Constructive: • x=0.n1…nmn1…nm… • x=0.(n1…nm) • x*10m-x=n1…nm • x=n1…nm/(10m-1)

  15. Theorem 2 Any geometric series: • S=q0+q1+q2+q3+… • where -1<q<1 • evaluates to S=1/(1-q) Proof • Proof idea • More formal proof • Definitions of limits and partial sums

  16. Z  Q Every integer is a rational number Proof : set the denominator to 1 • Book : page 127 Q The set of rational numbers is closed with respect to arithmetic operations +, -, *, / Partial proofs : textbook pages 121-131 Formal proof

  17. Irrational Numbers • So far all the examples were of rational numbers • How about some irrationals? •  • e • sqrt(2)

  18. Simple Exercises The sum of two even numbers is even. The product of two odd numbers is odd. direct proof.

  19. Divisibility a “divides” b or is b divisible by a (a|b ): b = ak for some integer k Also we say that b is multiple of a a is a factor of b b is divisor for a 5|15 because 15 = 35 n|0 because 0 = n0 1|n because n = 1n n|n because n = n1 A number p > 1 with no positive integer divisors other than 1 and itself is called a prime. Every other number greater than 1 is called composite. 2, 3, 5, 7, 11, and 13 are prime, 4, 6, 8, and 9 are composite.

  20. Simple Divisibility Facts 1. If a | b, then a | bc for all c. 2. If a | b and b | c, then a | c. 3. If a | b and a | c, then a | sb + tc for all s and t. 4. For all c ≠ 0, a | b if and only if ca | cb. Proof of (??) direct proof.

  21. Divisibility by a Prime Theorem. Any integer n > 1 is divisible by a prime number.

  22. Fundamental Theorem of Arithmetic Every integer, n>1, has a unique factorization into primes: p0≤ p1 ≤ ··· ≤ pk p0p1 ··· pk = n Example: 61394323221 = 3·3·3·7·11·11·37·37·37·53

  23. Prime Products Claim: Every integer > 1 is a product of primes. Proof:(by contradiction) Supposenot. Then set of non-products is nonempty. There is a smallest integer n > 1 that is not a product of primes. In particular, n is not prime. • So n = k·m for integers k, m where n > k,m >1. • Since k,m smaller than the leastnonproduct, both are prime products, eg., • k = p1 p2p94 • m = q1 q2   q214

  24. Prime Products Claim: Every integer > 1 is a product of primes. …So n = k m = p1 p2p94q1q2 q214 is a prime product, a contradiction.  The set of nonproducts > 1 must be empty. QED (The proof of the fundamental theorem will be given later.)

  25. The Quotient-Reminder Theorem For b> 0 and any a, there are unique numbers q : quotient(a,b), r: remainder(a,b), such that a = qb+ r and 0  r < b. We also say q = a div b r = a mod b. When b=2, this says that for any a, there is a unique q such that a=2q or a=2q+1. When b=3, this says that for any a, there is a unique q such that a=3q or a=3q+1 or a=3q+2.

  26. The Division Theorem For b> 0 and any a, there are unique numbers q : quotient(a,b), r: remainder(a,b), such that a = qb+ r and 0  r < b. Given any b, we can divide the integers into many blocks of b numbers. For any a, there is a unique “position” for a in this line. q = the block where a is in r = the offset in this block a (k+1)b kb 2b b -b 0 Clearly, given a and b, q and r are uniquely defined.

  27. The Square of an Odd Integer Idea 0: find counterexample. 32 = 9 = 8+1, 52 = 25 = 3x8+1 …… 1312 = 17161 = 2145x8 + 1, ……… Idea 1: prove that n2 – 1 is divisible by 8. Idea 2: consider (2k+1)2 Idea 3: Use quotient-remainder theorem.

  28. Contrapositive Proof Statement: If m2 is even, then m is even Contrapositive: If m is odd, then m2 is odd. Proof (the contrapositive): Since m is an odd number, m = 2l+1 for some natural number l. So m2 = (2l+1)2 = (2l)2 + 2(2l) + 1 So m2 is an odd number. Proof by contrapositive.

  29. Irrational Number Theorem:is irrational. Proof (by contradiction): • Suppose was rational. • Choose m, n integers without common prime factors (always possible) such that • Show that m and n are both even, thus having a common factor 2, • a contradiction!

  30. so m is even. Irrational Number Theorem:is irrational. Proof (by contradiction): Want to prove both m and n are even. so can assume so n is even. Proof by contradiction.

  31. Infinitude of the Primes Theorem. There are infinitely many prime numbers. Let p1, p2, …, pN be all the primes. Consider p1p2…pN + 1. Claim: if p divides a, then p does not divide a+1. Proof by contradiction.

  32. Floor and Ceiling Def: For any real number x, the floor of x, written x, is the unique integer n such that n  x < n + 1. It is the largest integer not exceeding x ( x). If k is an integer, what are x and x + 1/2 ? Is x + y = x + y? ( what if x = 0.6 and y = 0.7) For all real numbers x and all integers m, x + m = x + m For any integer n, n/2 is n/2 for even n and (n–1)/2 for odd n Def: For any real number x, the ceiling of x, written x, is the unique integer n such that n – 1 < x  n. What is n?

  33. Exercises Is it true that for all real numbers x and y: x – y = x - y x – 1 = x - 1 x + y = x + y x + 1 = x + 1 For positive integers n and d, n = d * q + r, where d = n / d and r = n – d * n / d with 0  r < d

  34. Greatest Common Divisors Given a and b, how to compute gcd(a,b)? Can try every number, but can we do it more efficiently? • Let’s say a>b. • If a=kb, then gcd(a,b)=b, and we are done. • Otherwise, by the Division Theorem, a = qb + r for r>0.

  35. Greatest Common Divisors • Let’s say a>b. • If a=kb, then gcd(a,b)=b, and we are done. • Otherwise, by the Division Theorem, a = qb + r for r>0. gcd(8,4) = 4 gcd(12,8) = 4 a=12, b=8 => 12 = 8 + 4 gcd(9,3) = 3 a=21, b=9 => 21 = 2x9 + 3 gcd(21,9) = 3 gcd(99,27) = 9 a=99, b=27 => 99 = 3x27 + 18 gcd(27,18) = 9 Euclid: gcd(a,b) = gcd(b,r)!

  36. Euclid’s GCD Algorithm a = qb + r Euclid: gcd(a,b) = gcd(b,r) gcd(a,b) if b = 0, then answer = a. else write a = qb + r answer = gcd(b,r)

  37. Example 1 gcd(a,b) if b = 0, then answer = a. else write a = qb + r answer = gcd(b,r) GCD(102, 70) 102 = 70 + 32 = GCD(70, 32) 70 = 2x32 + 6 = GCD(32, 6) 32 = 5x6 + 2 = GCD(6, 2) 6 = 3x2 + 0 = GCD(2, 0) Returnvalue:2. Example 3 Example 2 GCD(662, 414) 662 = 1x414 + 248 = GCD(414, 248) 414 = 1x248 + 166 = GCD(248, 166) 248 = 1x166 + 82 = GCD(166, 82) 166 = 2x82 + 2 = GCD(82, 2) 82 = 41x2 + 0 = GCD(2, 0) Returnvalue:2. GCD(252, 189) 252 = 1x189 + 63 = GCD(189, 63) 189 = 3x63 + 0 = GCD(63, 0) Returnvalue:63.

  38. Practice problems Study the Sections 3.1- 3.4 from your textbook. Be sure that you understand all the examples discussed in class and in textbook. Do the following problems from the textbook: Exercise 3.1 # 13, 16, 32, 36, 45 Exercise 3.2 # 15, 19, 21, 32, Exercise 3.3 # 13, 16, 25, 26, Exercise 3.4 # 4, 6, 8, 10, 18, 33

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