Number Theory: Understanding Patterns and Relationships in Natural Numbers, Study notes of Mathematics

Download Number Theory: Understanding Patterns and Relationships in Natural Numbers and more Study notes Mathematics in PDF only on Docsity! MATHEMATICS OF PATTERNS: NUMBER THEORY NOTES What Is Number Theory? Number theory is basically the investigation of patterns and relationships in the natural numbers (1, 2, 3, 4, . . . ). It first began as a specialized discipline of mathematics in the 17th century as a result of the work of Pierre Fermat and Karl Friedrich Gauss. But their work followed the work of the Greeks during 600 B.C to 400 A. D. While the Greeks appeared to do only geometry they investigated number theory by letting fixed length line segments represent numbers. Number theory is particularly interested in prime numbers, those numbers whose only factors are 1 and themselves. The su ( ) 2 21 3 5 7 2 1 m of consecutive odd numbers, starting with 1, is the square of the number of odd numbers added, or the sum of the first odd numbers is , or . . . Note that any odd number n n n n+ + + + + − = ( )2 1can be represented by The Greeks approached this by using a diagram, probably made with pebbles. Each "row-column" represents the next odd number, n− • • • • • • • • • • • • • • • counting the total pebbles gives you the sum of the first odd numbers These are also "square numbers", notice that 16 pebbles is made n • • • • • • • • • • of 4 rows and 4 columns, notice that 25 pebbles is made of 5 rows and 5 columns, and so forth. Example Problems: _____ 1. What is the 100th even number? _____ 2. What is the 994th odd number? 3, 6, 9, 12, 15, 18, _____ 3. What is the next number in the sequence? ____ ._ 4 … What is the 10th number in the sequence? _____ 5. Determine an expression for the th number in the sequence. _____ 6. What is the 200th number in the sequence? 1, 4, 7, 10, 1 n 3, 16, _____ 7. What is the next number in the sequence? _____ 8. What is the 10th number in the sequence? _____ 9. Determine an expression for the th number in the sequence. _____ 10. What is the n … 200th number in the sequence? 10, 17, 24, 31, 38, 45, _____ 11. What is the next number in the sequence? _____ 12. What is the 10th number in the sequence? _____ 13. Determine … an expression for the th number in the sequence. _____ 14. What is the 200th number in the sequence? n Triangular Numbers i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i 1 3 6 i i i i i i i i i i i i i i i i i i i i i i 10 15 21 __________ 11. What is the next triangular number in the sequence? __________ 12. What is the 10th triangular number in the sequence? __________ 13. Determine an expression for the th triangular number in the sequence. __________ 14. What is the 200th triangular number in the sequence? n Example Problems: 7 7 7 5 True or False _____1. 7 89 _____2. 8 300 _____3. 15 525 _____4. 5 10 _____5. If and , then _____6. If , then _____7. 5 7 8. Explain why the product of any two x y y x x y x y x y = < consecutive numbers is even. 9. Fill in the blanks to create an accurate definition: If ______ divides ______ , then is an even number. n Counting Divisors The will be denoted by . (read " of ") A is a number that h number of divisors of ( ) prime number fact as exactly two divisors. A of a number ia a representation of as a orizati producto o two n f D nn D n n n composite number Fundamental Theorem of Arit or more primes. A is a number with three or more divisors. Every number greater than one has a prime factorization. For any numbers hmetic unique and , m n m ( ) 3 1 2 3 if and only if (iff) every power of a prime factor of also divides . 8 24, number 8 2 , so 2 ,2 , and 2 all divide 24 The is represented by If is a p o rime Exampl :e Example: Example: f divisors o nu f n p n D n m n = ( )number of divisors omber then f the 1, k kp D p k= + e 5 divisors. D D= = + = 3 4 4 16 D D D D = ⋅ ( ) ( )4 4 3 is a prime number, the number of factors of 81 (which is 3 ) is 81 3 4 1 5 the divisors of 81 are 1, 3, 9, 27, and 81. There ar ( ) ( ) ( ) ( ) 3 3 3 3 What is 216 ? 216 2 3 So 216 2= ⋅ = ⋅ = n D n ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ Example Probl sem ( ) 2 4 4 2 10 2 5 4 28 2 7 6 278 2 D 139 4 ivisors 463 1 463 4 10,000 2 5 25 25,025 5 7 11 13 24 Summing Divisors 1 1 2 3 1 The of a number n is denoted by Note that if is prime then sum of the divisors ( ). ( ) 1 1If 1, then 1 + 1 k k k S n S n n nn n n n n n n n + − = + − > + + + …+ + = − TheExample sum of the divisors of 24 (24) The divisors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24 so (24) 1 2 3 4 6 8 12 24 6 S= 0= + + + + + + + = S 2 2 2 2 1 3 2 2 2 Determine (17 ) (17 ) 1 17 17 using the formula with 17, 2 17 1 17 1 4913 1 49121 17 17 307 17 1 17 1 16 16 so (289) (17 ) 1 17 17 1 17 289 307 S S n k S S + = + + = = − − − + + = = = = = − − = = + + = + + = 5 S S ( ) ( )( ) Example Example: ( ) ( ) 5 5 1 2 3 4 5 What is 3 ? The divisors of 3 are: 1, 3 , 3 , 3 , 3 , 3 So 3 =1+3+9+27+81+243=364 6 5 3 1 729 1 728 using the formula 3 364 3 1 3 1 2 S − − = = = = − − ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 1 1 3 1 1 2 1 3 1 2 3 1 1 3 1 1 Determine (60) (60) 2 3 5 2 3 5 using the formula with 2, 2 2 1 2 1 8 1 71 2 2 7 2 1 2 1 1 1 so (60) 2 3 5 2 3 5 7 4 6 168 1 2 3 SExample 4 5 6 10 12 S S S S S n k S S S S S + = ⋅ ⋅ = ⋅ ⋅ = = − − − + + = = = = = − − = ⋅ ⋅ = ⋅ ⋅ = ⋅ ⋅ = + + + + + + + 15 20 30 60 168+ + + + = Even Perfect Numbers Are these the only perfect num Some perfect numbers are 6, 28 bers? Are all perf , 496 ect numbers even? Determine whether the following numbers are perfect. 46 (46) 26, (46) 46, 46 is 48 (48) 76, 48 (48), 48 i deficient abundans Example t 5 P P P P = < = < 0 (50) 43, (50) 50, 50 is 52 (52) 46, (52) 52, 52 is 54 (54) 66, 54 (54), 5 defic 4 is ient deficient abundant None are perfect P P P P P P = < = < = < 1 1 2 1 4 1 Let's look at the prime factorization of our three perfect numbers (6, 28, 496) 6 2 3 28 2 7 Is there a pattern h 496 2 31 It looks like a power of two times a prime mi ere? = ⋅ = ⋅ = ⋅ 3 3 3 ght be a pattern. Further, it looks like the exponent of two is increasing as is the prime. What happened to 2 ? Let's try numbers that look like 2 where is a prime. 2 ( p n P p n p p= × × 11 88 92 Abundant 4 13 104 106 Abundant 2 17 136 134 Deficient 2 19 152 148 Deficient 4 23 152 176 Defic ) Classification ient 8 29 232 218 D ( eficient ) 14 P n n − − − − − 3So it doesn't look like we have a perfect number with as a factor but notice that goes from to as the prime changes from 13 to 17. Its almost like if there were a prime halfway b 2 ( etween ) 1 P n n + − − 33 and 17, then times that prime would be a perfect number. Notice that the numbers 3, 7, 15, 31 are all of the form 2 1 where each number is twice the prior plus one. So the next number that f s 2 it k − 6 7 that description is 63. 63 2 1 and 63 2 31 1 But 63 is composite so we probably should go to the next number. We find that the next number that fits the description is 127. 127 2 1 and 127 2 63 ⎡ ⎤= − = ⋅ +⎣ ⎦ = − = ⋅ + ( ) 6 6 7 2 127 8,128 (8128) (2 ) (127 1 And 127 is a prime n ) 2 1 128 127 128 16,256 (8128) 16,56 8,128 8,128 So 8,128 is a perfect umber! So our "fo numbe rmula" gives us r! S S S P × = = × = − × = × = = − = ⎡ ⎤⎣ ⎦ ( ) ( ) ( ) ( ) 1 1 If 2 1 is prime, then 2 2 1 is perfect A number of the form 2 2 1 Euclidean numb So we can make these statements . is called a er If and is denoted by . 2 1 is prime, then the Euclidean number k k k k k k k E − − − ⋅ − ⋅ − − ( ) ( )1 is perfect If is an even perfect number, it is of the form 2 2 1 where . 2 1 is pri .me k k k k E n − ⋅ − − Example Problems ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 5 1 5 1 5 4 5 5 1 1 1 1 1 0 1 1 What is ? 2 2 1 2 2 1 2 2 1 16 31 396 What is ? 2 2 1 2 2 1 2 2 1 1 1 1 k k k k k k E E E E E E − − − − = − = − = − = = = − = − = − = = Mersenne Primes A number of the form 2 1 Mersenne number If is prime, then is perfect. If , then (If is composite, then is called a and is composite is denoted . by But if is c . ) k k k k k kc M M E c k M M k M − is a Mersenne num1 ber. k k k M − ⋅ − − ( ) ( ) ( ) ( ) 89 89 1 8 1 8 89 88 89 If is prime then is perfect. 2 1 and it must be multiplied by 2 2 to yield a perfect number. So 2 2 1 is a perfect number. What odd numb 2 1 2 2 1 er k k k M − − ⋅ = − = − − − ( ) ( ) ( ) ( ) ( ) ( ) 12 12 1 1 3 can be multipied by 2 to produce a perfect number? A perfect number has the form So 2 2 1 = 4096 8191 =33,550,336 is a perfect numbe 2 . 2 1 r k k− − ⋅ − omposite th ien skM composite???????k Example Problems ( ) ( ) 8 1 9 What number can be multipied by to produce a perfect number? A perfect number has the form where 2 2 1 2