1 Elementary number theory
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1 1 Elementary number theory We assume the existence of the natural numbers and the integers N = {1, 2, 3,...} Z = {..., 3, 2, 1, 0, 1, 2, 3,...}, along with their most basic arithmetical and ordering properties. By this, we mean addition, subtraction and multiplication. For example, we assume the truth of statements such as For all integers a and b, both a + b and ab are integers. 0 < 1 For all integers a and b we have a + b = b + a. For any pair of integers a and b, exactly one of the following is true: a = b, a<bor a>b. WARNING: Until further notice, FRACTIONS ARE NOT ALLOWED. Definition 1. Let a and b be integers. We say a divides b if there exists an integer k such that ka = b. Notation 2. The notation a b means a divides b. Proposition 3. For any integers a, b and c, ifa b and a c then a (b + c). Proposition 4. For any integers a, b and c, if a b and a c then a (b c). Conjecture 5. For any integers a, b and c, ifa (b + c), then a b or a c. Proposition 6. For any integers a, b and c, ifa b then a bc. Proposition 7. For any integers a, b, d, x and y, ifd a and d b, then d (ax+by). Proposition 8. For any integers a, b and c, ifa b and b c then a c. Proposition The content of Proposition 9 should be interpreted as 2 does not divide 1.
2 1 Elementary number theory, II The setup is the same as in Worksheet I. You may assume all the Propositions from Worksheet I. Proposition 10. For any integer n we have n 0. Proposition 11. For any integers n and a we have n (a a). Proposition 12. For any integers n, a and b, if n (a b) then n (b a). Proposition 13. For any integers n, a, b and c, ifn (a b) and n (b c) then n (a c). Definition 14. Let a and b be integers and let n be a natural number. We say a is congruent to b modulo n if n (a b). Notation 15. We denote a is congruent to b modulo n by a b mod n. Proposition 16. For any integer a and natural number n, we have a a mod n. Proposition 17. For any integers a and b and any natural number n, ifa b mod n then b a mod n. Proposition 18. For any integers a, b and c and any natural number n, if a b mod n and b c mod n then a c mod n. Conjecture 19. For any integers a and b and any natural number n, ifa b mod n then ( a) ( b) modn. Conjecture 20. For any integer a and any natural number n, we have a a 2 mod n. Question 21. What is the usual word for a number a so that a 0mod2? What about those b so that b 1mod2?
3 1 Elementary number theory, III The setup is the same as in Worksheets I and II. You may assume all the Propositions from Worksheet I and from Worksheet II. Proposition 22. For any integers a, b, c and d and any natural number n, if a b mod n and c d mod n then a + c b + d mod n. Proposition 23. For any integers a, b, c and d and any natural number n, if a b mod n and c d mod n then ac bd mod n. Proposition 24. For any integers a, b, c and d and any natural number n, if a b mod n and c d mod n then a c b d mod n. Proposition 25. For any integers a and b and any natural number n, ifa b mod n then a 2 b 2 mod n. Conjecture 26. For any integers a and b and any natural number n, ifa b mod n then for all positive integers k, we have a k b k mod n. For the proof of Theorem 31 below, you may find the following lemma useful. Lemma 27. Let a and b be integers and let c be a natural number. If 0 apple a<c and 0 apple b<cthen c<a b<c, or equivalently a b <c. Definition 28. Let S be a set of integers. We say an element l in S is a least element (or a smallest element) ofs if for every s in S we have l apple s. Remark 29. We will assume that the following statement about the natural numbers is true: If S is any non-empty set of natural numbers then S has a least element. We ll refer to this statement as the Well-ordering Principle. Theorem 30. Let a be an integer and let b be a natural number. Then there exist integers q and r such that a = bq + r and 0 apple r<b. Theorem 31. Let a be an integer and let b be a natural number. If there exists integers q, q 0,r and r 0 such that a = bq+r, 0 apple r<band a = bq 0 +r 0, 0 apple r 0 <b then q = q 0 and r = r 0.
4 1 Elementary number theory, IV The setup is the same as in Worksheets I, II and III. You may assume all the Propositions from Worksheets I, II and III. [But, be warned that you are expected to know the proofs of these, and we will be revisiting them.] Definition 32. Let a and b be integers and suppose that not both of a and b are zero. An integer d is a greatest common divisor of a and b if the following two statements are true: 1. d a and d b; 2. for any integer c such that c a and c b we have c apple d. Proposition 33. Let a and b be integers which are not both zero. If d and d 0 are greatest common divisors of a and b, then d = d 0. Remark 34. Proposition 34 says that a greatest common divisor of a and b is unique (if it exists). The next result proves that the greatest common divisor of a and b exists. Theorem 35. Let a and b be integers such that not both are zero. Define a set The following statements hold: D = {am + bn m, n 2 Z,am+ bn > 0}. 1. D is a non-empty set of positive integers; 2. D has a smallest element d>0; 3. There are integers x and y so that d = ax + by; 4. d a ; 5. d b; 6. If c is any integer so that c a and c b then c d; 7. If c is any integer so that c a and c b then c apple d; 8. d is a greatest common divisor of a and b. Notation 36. The unique greatest common divisor of a and b is denoted gcd(a, b). Theorem 37. Let a and b be integers, not both zero. Then gcd(a, b) is equal to the smallest element of the set D = {am + bn > 0 m, n 2 Z}. Corollary 38. Suppose that a and b are integers, not both zero. integers x and y so that gcd(a, b) =ax + by. There are
5 1 Elementary number theory, V The setup is the same as in Worksheets I, II, III and IV. You may assume all the Propositions from Worksheets I, II, III and IV. [But, be warned that you are expected to know the proofs of these, and we will be revisiting them.] Definition 39. We say that integers a and b are relatively prime (or coprime) ifgcd(a, b) =1. Corollary 40. Let a and b be integers, not both zero. Then gcd(a, b) =1if and only if there exist integers x and y so that ax + by =1. Proposition 41. If n is an integer, then gcd(n, n + 1) = 1. Theorem 42. Let a, b and c be integers. If a bc and gcd(a, b) =1then a c. Proposition 43. Let a and b be integers such that at least one of a and b is not zero. If a = bq + r then gcd(a, b) = gcd(b, r). Example 44. Find gcd(835, 45), gcd(216, 57) and gcd(85, 31). Exercise 45. Describe a method suggested by Proposition 43 for finding the greatest common divisor of two integers. This method finds gcd(a, b) from a and b. Explainhowitalsofindsx and y so that ax + by = gcd(a, b). Theorem 46. Let a be an integer and n a natural number. If gcd(a, n) =1 then there exists an integer x such that ax 1modn. Example 47. Let a = 12 and n = 85. Use what we now know to find an integer x so that 12x 1 mod 85. Exercise 48. Find four examples of problems like Example 47. You should find a and n that are relatively prime, then find an integer x so that ax 1modn, for your choice(s) of a and n. Make sure that your examples are not too easy, but also not too hard. Question 49. Suppose that a and b are relatively prime integers. How many solutions (x, y) are there of the equation ax + by =1, for x and y integers? Is the solution unique? Are there at most five solutions? Are there finitely many? Are there infinitely many? Conjecture 50. Let a, b and c be integers with c 6= 0, and let n be a natural number. If ac bc mod n then a b mod n.
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