The sieve of Eratosthenes

1
The sieve of Eratosthenes

• Finding small prime numbers

2
Defining prime numbers

• A prime number has only two factors - the number
  itself and one. Examples are 2,3, 5.

3
Defining composite numbers

• Composite numbers have at least three factors.
• Examples are 4, 6, 9.

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What is the sieve of Eratosthenes?

• It is a way of separating composite numbers from
  prime numbers.

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Who was Eratosthenes?

• Eratosthenes was a Greek mathematician who
  devised a method of identifying prime numbers. He
  was a native of Cyrene (modern Libya in North
  Africa) who lived between 276 BC and 194 BC. He
  studied in Alexandria which was the centre of
  higher learning at this time.

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Using the sieve of Eratosthenes

• Since all the even numbers greater than 2
• 4, 6, 8, 10, 12 100 -
• are already gone since they are multiples of 2
• As 4 and 8 are also multiples of 2 we do not
  need to look at their multiples

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Using the sieve of Eratosthenes

• Multiples of 3 which exclude 3 itself are 6, 9,
  12, 15, 18 99 are gone
• Notice every second multiple of 3 is even - this
  is because these numbers are also multiples of 2
  and have already been eliminated
• You dont need to check for multiples of 6, 9
  etc. because they are multiples of 3

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Multiples of 5

• The next prime is 5
• All whole numbers ending with a 0 or a 5 are
  multiples of 5 and so multiples of 5 and 10 go in
  the one step

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Multiples of 7

• Composite multiples of 7 are
• 14 (2 ? 7),
• 21 (3 ? 7),
• 28 (4 ? 7)
• to the highest multiple up to 100 which is 14 ?
  7 98

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Primes from 10 and up to 20

• Lets look at the numbers the following way
• Prime numbers above 10 but less than 20 are-
• 11 is a prime number because its only factors
  are 1 and 11.
• 13, 17 and 19 are also primes

• Composite numbers are-
• 10, 12, 14, 16, 18, 20 all multiples of 2
• 12, 15, 18
• are multiples of 3
• 15 and 20
• are multiples of 5

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Prime numbers gt20 and up to and including 100

• All even numbers greater than 2 are gone
• All multiples of 3, 5 and 7 are gone
• For the primes between 10 and 20 - that is 11,
  13, 17 and 19 their multiples (other than the
  first) have been eliminated. So all the double
  numbers 22, 33, 44, 55 are gone

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Prime numbers gt20 and up to and including 100

• What is left?
• 23 and 29
• The multiples of 23 and 29 do not need to be
  eliminated. Why?

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Multiples of 23 and 29

• 23 x 2 46 and 23 x 4 96 went with all even
  numbers (multiples of 2) as does
• 58 (29 x 2)
• 29 x 3 87
• went as a multiple of 3

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Prime numbers gt20 and up to and including 100

• Between 30 and 40
• you get 31 and 37
• Between 40 and 50
• you get 41, 43 and 47
• From the last slide you can see that their
  multiples were of course eliminated with the
  multiples of 2 and 3

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Looking for Patterns

• 53 is the first prime above 50. There cant be
  any multiples of these since
• 2 ? 53 106 a number greater than 100.

• From 40 to 100 are primes are
• 41 43 47
• 53 59
• 61 67
• 71 73 79
• 83 89
• 91 97

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The final digit is important

• The first pattern you probably noticed is that
• the last digit of each number is odd - but never
  a 5
• Probably you also noticed that the final digit of
  each number must be a
• 1, 3, 7 or 9

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Patterns in primes 20 to 100

• 23 29
• 31 37
• 41 43 47 53 59
• 61 67 71 73 79
• 83 89
• 91 97

• In each decade (group of 10 numbers) above 20
  there is either 2 or 3 primes
• Each final digit is
• 1, 3, 7 or 9
• The pattern in the 3rd decade (20 - 29) repeats
  in the 6th decade (50 - 59) and the 9th decade
  (80 - 89)
• The pattern of the 4th decade repeats in the 7th
  and the 10th decades
• The pattern of the 5th decade repeats in the 8th

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Predicting from observed patterns

• Since you have observed a pattern which recurs
  each 3 decades, which decade pattern would you
  look to in order to predict the prime numbers in
  11th decade (100 to 110)?
• Work it out for yourself.

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Your choice was

• Of course you chose the 8th decade
• 71 73 79
• and

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• the 11th decade it is
• 101 103 109