1 Elementary number theory
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1 Math Introduction to Advanced Mathematics Spring 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, but not division. 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 < b or 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, if a 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, if a (b + c), then a b or a c. Proposition 6. For any integers a, b and c, if a b then a bc. Proposition 7. For any integers a, b, d, x and y, if d a and d b, then d (ax+by). Proposition 8. For any integers a, b and c, if a b and b c then a c. Proposition 9. For any integers c and d so that c, d 0, if c d then c d. Proposition The content of Proposition 10 should be interpreted as 2 does not divide 1. Proposition 11. For all integers n we have n 0.
2 Proposition 12. For all integers n and a we have n (a a). Proposition 13. For all integers n, a and b, if n (a b) then n (b a). Proposition 14. For all integers n, a, b and c, if n (a b) and n (b c) then n (a c). Definition 15. 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 16. We denote a is congruent to b modulo n by a b mod n. Proposition 17. For all integers a and all natural numbers n, we have a a mod n. Proposition 18. For all integers a and b and all natural numbers n, if a b mod n then b a mod n. Proposition 19. For all integers a, b and c and all natural numbers n, if a b mod n and b c mod n then a c mod n. Conjecture 20. For all integers a and b and all natural numbers n, if a b mod n then ( a) ( b) mod n. Conjecture 21. For all integers a and all natural numbers n, we have a a 2 mod n. Question 22. What is the usual word for a number a so that a 0 mod 2? What about those b so that b 1 mod 2? 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 a + c b + d 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 ac bd mod n. Proposition 25. 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 26. For any integers a and b and any natural number n, if a b mod n then a 2 b 2 mod n. Conjecture 27. For any integers a and b and any natural number n, if a b mod n then for all positive integers k, we have a k b k mod n. For the proof of Theorem 32 below, you may find the following lemma useful. Lemma 28. Let a and b be integers and let c be a natural number. If 0 a < c and 0 b < c then c < a b < c, or equivalently a b < c.
3 Definition 29. Let S be a set of integers. We say an element l in S is a least element (or a smallest element) of S if for every s in S we have l s. Remark 30. 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 31. 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 r < b. Theorem 32. Let a be an integer and let b be a natural number. If there exists integers q, q, r and r such that a = bq+r, 0 r < b and a = bq +r, 0 r < b then q = q and r = r. Corollary 33. Let n be a natural number, and consider the relation on the integers defined by a b if a b mod n. There are exactly n equivalence classes which are [i] where 0 i < n is an integer. Definition 34. 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 d. Proposition 35. Let a and b be integers which are not both zero. If d and d are greatest common divisors of a and b, then d = d. Remark 36. Proposition 35 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 37. Let a and b be integers such that not both are zero. Define a set The following statements hold: D = {am + bn m, n 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;
4 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 d; 8. d is a greatest common divisor of a and b. Notation 38. The unique greatest common divisor of a and b is denoted gcd(a, b). Theorem 39. 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 Z}. Corollary 40. Suppose that a and b are integers, not both zero. integers x and y so that gcd(a, b) = ax + by. There are Definition 41. We say that integers a and b are relatively prime (or coprime) if gcd(a, b) = 1. Corollary 42. Let a and b be integers, not both zero. Then gcd(a, b) = 1 if and only if there exist integers x and y so that ax + by = 1. Proposition 43. If n is an integer, then gcd(n, n + 1) = 1. Theorem 44. Let a, b and c be integers. If a bc and gcd(a, b) = 1 then a c. Proposition 45. 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 46. Find gcd(835, 45), gcd(216, 57) and gcd(85, 31). Exercise 47. Describe a method suggested by Proposition 45 for finding the greatest common divisor of two integers. This method finds gcd(a, b) from a and b. Explain how it also finds x and y so that ax + by = gcd(a, b). Theorem 48. Let a be an integer and n a natural number. If gcd(a, n) = 1 then there exists an integer x such that ax 1 mod n. Example 49. Let a = 12 and n = 85. Use what we now know to find an integer x so that 12x 1 mod 85. Exercise 50. Find four examples of problems like Example 49. You should find a and n that are relatively prime, then find an integer x so that ax 1 mod n, for your choice(s) of a and n. Make sure that your examples are not too easy, but also not too hard.
5 Question 51. 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 52. Let a, b and c be integers with c 0, and let n be a natural number. If ac bc mod n then a b mod n. Definition 53. We say that a natural number n > 1 is composite if there are natural numbers u and v with u, v < n and n = uv. Definition 54. We say that a natural number p > 1 is prime if there do not exist natural numbers u, v with u, v < p so that p = uv. Theorem 55. Let n > 1 be a natural number. Then there exists a prime p such that p n. Lemma 56. Suppose that p is a prime number and n is a natural number. If n p then n = 1 or n = p. Proposition 57. Suppose that p is a prime and that a is a natural number. Then either gcd(a, p) = 1 or gcd(a, p) = p. The case gcd(a, p) = p happens if and only if p a. Theorem 58. Suppose that p is a prime, and that a and b are natural numbers. If p ab then p a or p b. Remark 59. Theorem 58 is called Euclid s Lemma. Theorem 60. There are infinitely many prime numbers.
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