Full Binary Trees

1. Full Binary Trees Definition: The heighth(T) of a full binary tree T is defined recursively as follows: BASIS STEP: The height of a full binary tree T consisting of only a root r is h(T) = 0. RECURSIVE STEP: If T1 and T2are full binary trees, then the full binary tree T = T1∙T2has height h(T) = 1 + max(h(T1),h(T2)).

2. Full Binary Trees The number of vertices n(T) of a full binary tree T satisfies the following recursive formula: BASIS STEP: The number of vertices of a full binary tree T consisting of only a root r is n(T) = 1. RECURSIVE STEP: If T1 and T2are full binary trees, then the full binary tree T = T1∙T2has the number of vertices n(T) = 1 + n(T1) + n(T2).

3. Recursive Algorithms Definition: An algorithm is called recursive if it solves a problem by reducing it to an instance of the same problem with smaller input. For the algorithm to terminate, the instance of the problem must eventually be reduced to some initial case for which the solution is known.

4. Recursively Defined Functions Example: Give a recursive definition of the factorial function n!: Solution: f(0) = 1 f(n + 1) = (n + 1) * f(n)

5. Recursive Factorial Algorithm Example: Give a recursive algorithm for computing n!, where n is a nonnegative integer. Solution: Use the recursive definition of the factorial function. procedure factorial(n:nonnegative integer) if n = 0 then return 1 else return n∙factorial(n − 1) {output is n!}

6. Recursive Binary Search Algorithm Example: Construct a recursive version of a binary search algorithm. Solution: Assume we have a1,a2,…, an, an increasing sequence of integers. Initially i is 1 and j is n. We are searching for x. procedure binary search(i, j, x : integers, 1≤ i ≤ j ≤n) m := ⌊(i + j)/2⌋ if x = amthen return m else if (x < am and i < m) then return binary search(i,m−1,x) else if (x > am and j >m) then return binary search(m+1,j,x) else return 0 {output is location of x in a1, a2,…,anif it appears, otherwise 0}

7. Proving Recursive Algorithms Correct Both mathematicaland strong induction are useful techniques to show that recursive algorithms always produce the correct output. Example: Prove that the algorithm for computing the powers of real numbers is correct. Solution: Use mathematical induction on the exponent n. BASIS STEP: a0 = 1 for every nonzero real number a, and power(a, 0) = 1. INDUCTIVE STEP: The inductive hypothesis is that power(a,k) = ak, for all a≠ 0. Assuming the inductive hypothesis, the algorithm correctly computes ak+1, Sincepower(a, k + 1) =a ∙ power (a, k) = a ∙ ak =ak+1 . procedurepower(a:nonzeroreal number, n:nonnegative integer) if n = 0 then return 1 else return a∙ power (a, n − 1) {output is an} −2 .

8. Chapter Motivation • Number theory is the part of mathematics devoted to the study of the integers and their properties. • Key ideas in number theory include divisibility and the primality of integers. • Representations of integers, including binary and hexadecimal representations, are part of number theory.

9. Chapter Summary • Divisibility and Modular Arithmetic • Integer Representations and Algorithms • Primes and Greatest Common Divisors

10. Divisibility and Modular Arithmetic Section 4.1

11. Section Summary • Division • Division Algorithm • Modular Arithmetic

12. Division Definition: If a and b are integers with a ≠ 0, then adividesb if there exists an integer c such that b = ac. • When a divides b we say that a is a factor or divisor of b and that b is a multiple of a. • The notation a | b denotes that a divides b. • If a | b, then b/a is an integer. • If a does not divide b, we write a∤b. Example: Determine whether 3 | 7 and whether 3 | 12.

13. Properties of Divisibility Theorem 1: Let a, b, and c be integers, where a≠0. • If a | b and a | c, then a | (b + c); • If a | b, then a | bc for all integers c; • If a | b and b | c, then a | c. Proof: (i) Suppose a | b and a | c, then it follows that there are integers s and t with b = as and c = at. Hence b + c = as + at = a(s + t). Hence, a | (b + c) (Exercises 3 and 4 ask for proofs of parts (ii) and (iii).) Corollary: If a, b, and c be integers, where a≠ 0, such that a | b and a | c, then a | mb + nc whenever m and n are integers. Can you show how it follows easily from from (ii) and (i) of Theorem 1?

14. Division Algorithm • When an integer is divided by a positive integer, there is a quotient and a remainder. This is traditionally called the “Division Algorithm,” but is really a theorem. Division Algorithm: If a is an integer and d a positive integer, then there are unique integers q and r, with 0 ≤ r < d, such that a = dq +). • d is called the divisor. • a is called the dividend. • q is called the quotient. • r is called the remainder. Examples: • What are the quotient and remainder when 101 is divided by 11? Solution: The quotient when 101 is divided by 11 is 9 = 101 div11, and the remainder is 2 = 101mod11. • What are the quotient and remainder when −11 is divided by 3? Solution: The quotient when −11 is divided by 3 is −4 = −11 div3, and the remainder is 1 = −11mod3. • Definitions of Functions div and mod • q = a div d • r = a mod d

15. The Relationship between (mod m) and modm Notations • The use of “mod” in a ≡b (mod m)and a mod m = b are different. • a ≡b (mod m) is a relation on the set of integers. • In a mod m = b, the notation mod denotes a function. • The relationship between these notations is made clear in this theorem. • Theorem 3: Let a and b be integers, and let m be a positive integer. Then a ≡ b (mod m) if and only if a mod m = b mod m.

16. Computing the modm Function of Products and Sums We use the following corollary to Theorem 5 to compute the remainder of the product or sum of two integers when divided by m from the remainders when each is divided by m. Corollary: Let m be a positive integer and let aandb be integers. Then (a + b) (mod m) = ((a mod m) + (b mod m)) mod m and ab mod m= ((amod m)(bmod m)) mod m. (proof in text)

17. Arithmetic Modulo m Definitions: Let Zm be the set of nonnegative integers less than m: {0,1, …., m−1} • The operation +m is defined as a +m b = (a + b) modm. This is addition modulo m. • The operation ∙m is defined as a ∙mb = (a + b) modm. This is multiplication modulo m. • Using these operations is said to be doing arithmetic modulo m. Example: Find 7 +11 9 and 7 ∙11 9. Solution: Using the definitions above: 7 +11 9 = (7 + 9) mod 11 = 16 mod 11 = 5 7 ∙11 9 = (7 ∙ 9) mod 11 = 63 mod 11 = 8

18. Arithmetic Modulo m The operations +m and ∙m satisfy many of the same properties as ordinary addition and multiplication. • Closure: If a and b belong to Zm , thena +m b and a∙m b belong to Zm . • Associativity: If a, b, and c belong to Zm , then (a +m b) +m c = a +m (b +m c) and (a∙m b)∙m c = a∙m (b∙m c). • Commutativity: If a and b belong to Zm , then a +m b = b +m a and a∙m b = b∙m a. • Identity elements: The elements 0 and 1 are identity elements for addition and multiplication modulo m, respectively. • If a belongs to Zm , then a +m 0 = a and a∙m 1 = a. continued→

19. Arithmetic Modulo m Additive inverses: If a ≠ 0 belongs to Zm , then m − a is the additive inverse of a modulo m and 0 is its own additive inverse. a +m (m− a ) = 0 and 0 +m 0 = 0 Distributivity: If a, b, and c belong to Zm , then a∙m (b +m c) = (a∙m b) +m (a ∙m c) and (a+m b)∙m c = (a ∙m c) +m (b∙m c). Multiplicatative inverses have not been included since they do not always exist. For example, there is no multiplicative inverse of 2 modulo 6.

20. Integer Representations and Algorithms Section 4.2

21. Section Summary • Integer Representations • Base b Expansions • Binary Expansions • Octal Expansions • Hexadecimal Expansions • Base Conversion Algorithm • Algorithms for Integer Operations

22. Representations of Integers • In the modern world, we use decimal, or base10,notation to represent integers. For example when we write 965, we mean 9∙102 + 6∙101 + 5∙100 . • We can represent numbers using any base b, where b is a positive integer greater than 1. • The bases b = 2 (binary), b = 8 (octal) , and b= 16 (hexadecimal) are important for computing and communications • The ancient Mayans used base 20 and the ancient Babylonians used base 60.

23. Base b Representations • We can use positive integer b greater than 1 as a base, because of this theorem: Theorem 1: Let b be a positive integer greater than 1. Then if n is a positive integer, it can be expressed uniquely in the form: n = akbk + ak-1bk-1+ …. + a1b + a0 where k is a nonnegative integer, a0,a1,…. ak are nonnegative integers less than b, and ak≠ 0. The aj, j = 0,…,k are called the base-b digits of the representation. The representation of n given in Theorem 1 is called the base b expansion of n and is denoted by (akak-1….a1a0)b. We usually omit the subscript 10 for base 10 expansions.

24. Binary Expansions Most computers represent integers and do arithmetic with binary (base 2) expansions of integers. In these expansions, the only digits used are 0 and 1. Example: What is the decimal expansion of the integer that has (101011111)2 as its binary expansion? Solution: (1 0101 1111)2 = 1∙28 + 0∙27 + 1∙26 + 0∙25 + 1∙24 + 1∙23 + 1∙22 + 1∙21 + 1∙20 =351.

25. Binary Expansions Example: What is the decimal expansion of the integer that has (11011)2 as its binary expansion? Solution: (11011)2 = 1 ∙24 + 1∙23 + 0∙22 + 1∙21 + 1∙20 =27.

26. Octal Expansions The octal expansion (base 8) uses the digits {0,1, 2, 3, 4, 5, 6, 7}. Example: What is the decimal expansion of the number with octal expansion (7016)8 ? Solution: 7∙83 + 0∙82 + 1∙81 + 6∙80 =3598

27. Octal Expansions Example: What is the decimal expansion of the number with octal expansion (111)8 ? Solution: 1∙82 + 1∙81 + 1∙80 = 64 + 8 + 1 = 73

28. Hexadecimal Expansions The hexadecimal expansion needs 16 digits, but our decimal system provides only 10. So letters are used for the additional symbols. The hexadecimal system uses the digits {0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F}. The letters A through F represent the decimal numbers 10 through 15. Example: What is the decimal expansion of the number with hexadecimal expansion (2AE0B)16? Solution: 2∙164 + 10∙163 + 14∙162 + 0∙161 + 11∙160 =175627

29. Hexadecimal Expansions Example: What is the decimal expansion of the number with hexadecimal expansion (1E5)16? Solution: 1∙162+ 14∙161 + 5∙160 = 256 + 224 + 5 = 485

30. Hexadecimal Expansions Solution: 1∙162+ 14∙161 + 5∙160 = 256 + 224 + 5 = 485

31. Octal Expansions Example: What is the decimal expansion of the number with octal expansion (111)8 ? Solution: 1∙82 + 1∙81 + 1∙80 = 64 + 8 + 1 = 73

32. Base Conversion To construct the base b expansion of an integer n: • Divide n by b to obtain a quotient and remainder. n = bq0 + a0 0 ≤ a0 ≤b • The remainder, a0 , is the rightmost digit in the base b expansion of n. Next, divide q0 by b. q0 = bq1 + a1 0 ≤ a1 ≤b • The remainder, a1, is the second digit from the right in the base b expansion of n. • Continue by successively dividing the quotients by b, obtaining the additional base b digits as the remainder. The process terminates when the quotient is 0. continued→

33. Algorithm: Constructing Base b Expansions procedurebase b expansion(n, b: positive integers with b > 1) quotient := n k := 0 while (quotient ≠ 0) ak := quotientmodb quotient:= quotientdivb k := k + 1 return(ak-1 ,…, a1,a0) {(ak-1 … a1a0)b is base b expansion of n} • q represents the quotient obtained by successive divisions by b, starting with q = n. • The digits in the base b expansion are the remainders of the division given by qmodb. • The algorithm terminates when q = 0 is reached.

34. Base Conversion Example: Find the octal expansion of (12345)10 Solution: Successively dividing by 8 gives: • 12345 = 8 ∙ 1543 + 1 • 1543 = 8 ∙ 192 + 7 • 192 = 8 ∙ 24 + 0 • 24 = 8 ∙ 3 + 0 • 3 = 8 ∙ 0 + 3 The remainders are the digits from right to left yielding (30071)8.

35. Comparison of Hexadecimal, Octal, and Binary Representations Initial 0s are not shown Each octal digit corresponds to a block of 3 binary digits. Each hexadecimal digit corresponds to a block of 4 binary digits. So, conversion between binary, octal, and hexadecimal is easy.

36. Conversion Between Binary, Octal, and Hexadecimal Expansions Example: Find the octal and hexadecimal expansions of (11 1110 1011 1100)2. Solution: • To convert to octal, we group the digits into blocks of three (011 111 010 111 100)2, adding initial 0s as needed. The blocks from left to right correspond to the digits 3, 7, 2, 7, and 4. Hence, the solution is (37274)8. • To convert to hexadecimal, we group the digits into blocks of four (0011 1110 1011 1100)2, adding initial 0s as needed. The blocks from left to right correspond to the digits 3, E, B, and C. Hence, the solution is (3EBC)16.

37. Binary Addition of Integers Algorithms for performing operations with integers using their binary expansions are important as computer chips work with binary numbers. Each digit is called a bit. The number of additions of bits used by the algorithm to add two n-bit integers is O(n). procedureadd(a, b: positive integers) {the binary expansions of a and b are (an-1, an-2, …, a0)2 and (bn-1, bn-2, …, b0)2, respectively} c := 0 for j := 0 to n− 1 d := ⌊(aj + bj + c)/2⌋ sj:= aj + bj + c − 2d c := d sn:= c return(s0 ,s1, …, sn){the binary expansion of the sum is (sn, sn-1, …, s0)2}

38. Binary Multiplication of Integers • Algorithm for computing the product of two n bit integers. • The number of additions of bits used by the algorithm to multiply two n-bit integers is O(n2). proceduremultiply(a, b: positive integers) {the binary expansions of a and b are (an-1, an-2, …, a0)2 and (bn-1, bn-2, …, b0)2, respectively} for j := 0 to n− 1 if bj = 1 then cj = a shifted j places else cj:= 0 {co,c1,…, cn-1 are the partial products} p := 0 for j := 0 to n− 1 p := p+ cj returnp {p is the value of ab}

39. Binary Modular Exponentiation • In cryptography, it is important to be able to find bnmodm efficiently, where b, n, and m are large integers. • Use the binary expansion of n, n = (ak-1, …, a1, ao)2 , to compute bn. Note that: • Therefore, to compute bn, we need only compute the values of b, b2, (b2)2 = b4, (b4)2 = b8 , …, and the multiply the terms in this list, where aj = 1. Example: Compute 311using this method. Solution: Note that 11 = (1011)2 so that 311= 38 32 31= ((32)2 )2 32 31 = (92 )2 ∙ 9 ∙ 3 = (81)2 ∙ 9 ∙ 3 = 6561∙ 9 ∙ 3=117,147. continued→

40. Binary Modular Exponentiation Algorithm • The algorithm successively finds bmodm,b2modm, b4modm, …, modm, and multiplies together the terms where aj = 1. • O((log m )2 log n) bit operations are used to find bnmodm. proceduremodular exponentiation(b: integer, n = (ak-1ak-2…a1a0)2 , m: positive integers) x := 1 power := bmodm for i := 0 to k− 1 if ai= 1 then x:= (x∙ power ) modm power := (power∙ power ) modm returnx {x equals bnmodm }

41. Primes and Greatest Common Divisors Section 4.3

42. Section Summary • Prime Numbers and their Properties • Greatest Common Divisors and Least Common Multiples • The Euclidian Algorithm • gcds as Linear Combinations

43. Primes Definition: A positive integer p greater than 1 is called prime if the only positive factors of p are 1 and p. A positive integer that is greater than 1 and is not prime is called composite. Example: The integer 7 is prime because its only positive factors are 1 and 7, but 9 is composite because it is divisible by 3.

44. The Fundamental Theorem of Arithmetic Theorem: Every positive integer greater than 1 can be written uniquely as a prime or as the product of two or more primes where the prime factors are written in order of nondecreasing size. Examples: • 100 = 2 ∙ 2 ∙ 5 ∙ 5 = 22 ∙ 52 • 641 = 641 • 999 = 3 ∙ 3 ∙ 3 ∙ 37 = 33 ∙ 37 • 1024 = 2 ∙ 2 ∙ 2 ∙ 2 ∙ 2 ∙ 2 ∙ 2 ∙ 2 ∙ 2 ∙ 2 = 210

45. The Sieve of Erastosthenes Erastothenes (276-194 B.C.) • The Sieve of Erastosthenes can be used to find all primes not exceeding a specified positive integer. For example, begin with the list of integers between 1 and 100. • Delete all the integers, other than 2, divisible by 2. • Delete all the integers, other than 3, divisible by 3. • Next, delete all the integers, other than 5, divisible by 5. • Next, delete all the integers, other than 7, divisible by 7. • Since all the remaining integers are not divisible by any of the previous integers, other than 1, the primes are: {2, 3, 7,11,19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97} continued →

46. Greatest Common Divisor Definition: Let a and b be integers, not both zero. The largest integer d such that d | a and also d | b is called the greatest common divisor of a and b. The greatest common divisor of a and b is denoted by gcd(a, b). One can find greatest common divisors of small numbers by inspection. Example: What is the greatest common divisor of 24 and 36? Solution: gcd(24, 26) = 12 Example: What is the greatest common divisor of 17 and 22? Solution: gcd(17, 22) = 1

47. Greatest Common Divisor Definition: The integers a and b are relatively prime if their greatest common divisor is 1. Example: 17 and 22

48. Greatest Common Divisor Definition: The integers a1, a2, …, an are pairwiserelatively prime if gcd(ai, aj)= 1 whenever 1 ≤ i<j ≤n. Example: Determine whether the integers 10, 17, and 21 are pairwise relatively prime. Solution: Because gcd(10, 17) = 1, gcd(10, 21) = 1, and gcd(17,21) = 1, 10, 17, and 21 are pairwise relatively prime.

49. Greatest Common Divisor Example: Determine whether the integers 10, 19, and 24 are pairwise relatively prime. Solution: Because gcd(10, 24) = 2, 10, 19, and 24 are not pairwise relatively prime.

50. Least Common Multiple Definition: The least common multiple of the positive integers a and b is the smallest positive integer that is divisible by both a and b. It is denoted by lcm(a,b). The least common multiple can also be computed from the prime factorizations. This number is divided by both a and b and no smaller number is divided by a and b. Example: lcm(233572, 2433) = 2max(3,4) 3max(5,3) 7max(2,0) = 24 35 72 The greatest common divisor and the least common multiple of two integers are related by: Theorem 5: Let a and b be positive integers. Then ab = gcd(a,b) ∙ lcm(a,b) (proof is Exercise 31)