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Set Of Primes Is Infinite - Number Theory
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Set Of Primes Is Infinite - Number Theory

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this is a presentation on a a number theory topic concerning primes, it discusses three topics, the sieve of Eratosthenes, the euclids proof that primes is infinite, and solving for tau (n) primes.

this is a presentation on a a number theory topic concerning primes, it discusses three topics, the sieve of Eratosthenes, the euclids proof that primes is infinite, and solving for tau (n) primes.

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

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