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- 1. <ul><li>In mathematics , the Sieve of Eratosthenes is a simple, ancient algorithm for finding all prime numbers up to a specified integer. </li></ul><ul><li>It was created by Eratosthenes , an ancient Greek mathematician . </li></ul>
- 2. <ul><li>Consider a contiguous list of numbers from two to some maximum. </li></ul><ul><li>Strike off all multiples of 2 greater than 2 from the list. </li></ul><ul><li>The next lowest, uncrossed off number in the list is a prime number. </li></ul><ul><li>Strike off all multiples of this number from the list. The crossing-off of multiples can be started at the square of the number, as lower multiples have already been crossed out in previous steps. </li></ul><ul><li>Repeat steps 3 and 4 until you reach a number greater than the square root of the highest number in the list; all the numbers remaining in the list are prime. </li></ul>
- 3. <ul><li>Suppose we want to determine all primes less than 100. </li></ul><ul><li>First we write the integers from 2 to 100. </li></ul><ul><li>we know that 2 is prime; so we encircle it, and cross out the remaining even numbers on the list. </li></ul><ul><li>Now, the lowest number is 3; so we cross out again every 3 rd number thereafter. </li></ul><ul><li>We continue the same process until we reach the last number that hasn’t been crossed out. </li></ul>
- 4. <ul><li>In order to find all the primes up to n, we need only to seive out multiples of primes ≤ √n. </li></ul><ul><li>To find the primes up to 100, we need only to cross out multiples of 2,3,5 and 7. </li></ul><ul><li>The operation of the Sieve of Eratosthenes, suggests that primes becomes rarer as the integer goes up. </li></ul><ul><li>For example, in finding for the prime numbers bet. 100-150. </li></ul><ul><li>It consists of crossing out the multiple of 5 primes ( 2,3,5,7,11) not exceeding √150 ≈ 12.2 </li></ul>
- 5. <ul><li>The sieve suggests that at some point, the primes will become inexixtent. This is in contrary to what euclid said, “that primes are infinite”. </li></ul><ul><li>Euclid’s proof that primes is infinite </li></ul><ul><li>Q= p 1 p 2 ..........p k + 1 </li></ul><ul><li>Where: </li></ul><ul><li>Q – either prime or has a prime factor </li></ul><ul><li>p 1 p 2 ..........p k – prime numbers. </li></ul>
- 6. <ul><li>Find all primes between 5- and 100. </li></ul><ul><li>Find all primes between 100 and 150. </li></ul><ul><li>Find all primes between 1000 and 1025. </li></ul><ul><li>Find ( P ,n ) for n = 11 , 12 , . . . , 20 </li></ul><ul><li>Find ( P , n ) for n = 21 , 22 , . . . , 30. </li></ul>
- 7. <ul><li>Τ (n) </li></ul><ul><li>So, going back to the previous topics on the tau τ (n). </li></ul><ul><li>Suppose, </li></ul><ul><li>n = p 1 k 1 p 2 k 2 . . . . . P t k t , </li></ul><ul><li>Where , 0 ≤ j 1 ≤ k 1 for i = 1,2 . . . , t </li></ul><ul><li>Example: n= 63 = 3 2 7, then the divisors are </li></ul><ul><li>3 0 7 0 3 1 7 0 3 2 7 0 </li></ul><ul><li> 3 0 7 1 3 1 7 1 3 2 7 1 </li></ul>
- 8. <ul><li>When wrtting for the divisors of a certain imteger, say 63, we find it convenient to organize them into a rectangular array. </li></ul><ul><li>This array reminds us of a multiplication table, because it is indeed. </li></ul><ul><li> 1 3 9 </li></ul><ul><li>1 1 3 9 </li></ul><ul><li> 7 7 21 63 </li></ul>
- 9. <ul><li>As shown in the table, we have written the positive divisors of 63 ( 9 , 7 ). </li></ul><ul><li>Each entry in the table is the product of a divisor of 9 and one of 7. Thus in this case </li></ul><ul><li>Τ ( n) = Τ (a ) Τ ( b) = a ∙ b = ab </li></ul><ul><li>Τ ( 63) = Τ (9 ) Τ ( 7) = 3 ∙ 2 = 6 </li></ul><ul><li>This suggests that we can prove that Τ ( n) = Τ (a ) Τ ( b) = a ∙ b = ab by showing the divisors of ab are just all products of a divisor of a with one of b. </li></ul>
- 10. <ul><li>Theorem 2.11. if a and b are relatively prime positive integers, then τ (ab) = τ (a) τ (b). </li></ul><ul><li>This is the theorem that served as the basis for our computations used in finding primes. </li></ul><ul><li>Theorem 2.12. if a and b are relatively prime positive integers, then the set of positive dicvisors of ab consists exactly of all products de, where d is a positive divisor of a and e is a positive divisor of b. Furthermore, these products are all distinct. </li></ul>
- 11. <ul><li>Find τ ( n ) by means of the formula presented in this discussion: </li></ul><ul><li>n = 75 n = 45 n = 30 </li></ul><ul><li>Make a multiplication table of products of positive divisors of a times positive divisor of b . </li></ul><ul><li>a = 8 , b = 15 </li></ul><ul><li>a = 28 , b = 21 </li></ul>
- 12. <ul><li>Any questions?????????? </li></ul><ul><li>Clarifications????????? </li></ul><ul><li>Thank you and Good Day!!!!!!!!!!!!! </li></ul>

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