Learning Session 4: Prime Number Investigation
Prime numbers have fascinated mathematicians from many cultures for thousands of years. Nowadays they are important because the best commercial and military ciphers depend on their properties. In fact large prime numbers are used in the process that provides email security.
1.
Follow the instructions below which show how to use the Sieve of Eratosthenes to identify prime numbers to 100. Remember that a prime number can be divided only by itself and one and 2 is the smallest prime.
1 2 3 4 5 6 7 8 9 10
11 12 13 14 15 16 17 18 19 20
21 22 23 24 25 26 27 28 29 30
31 32 33 34 35 36 37 38 39 40
41 42 43 44 45 46 47 48 49 50
51 52 53 54 55 56 57 58 59 60
61 62 63 64 65 66 67 68 69 70
71 72 73 74 75 76 77 78 79 80
81 82 83 84 85 86 87 88 89 90
91 92 93 94 95 96 97 98 99 100
The sieve gives you a simple method for finding prime numbers.
Cross 1 out.
Then starting from 2, circle 2 but cross out every multiple of 2 from your sieve.
Starting with 3, circle 3, but cross out every multiple of 3 from your sieve.
Starting with 5, circle 5, but cross out every multiple of 5 from your sieve.
Starting with 7, circle 7 but cross out every multiple of it.
Do the same with all the primes that you know already. The numbers that are crossed are not primes, because they are multiples of other numbers. The numbers that are circled are primes. They should have no divisors apart from themselves and 1.
Make a list of your primes.
2.
Do you think the whole numbers greater than 5 can each be written as a sum of 3 primes?
Write down your answer: yes or no.
Now try to write each number from 6 to 30 as a sum of 3 primes. Here are the first three:
6 = 2+2+2
7 = 2+2+3
8 = 2+3+3
Record your findings and anything you think might be of interest. Do you want to change your answer to the question?
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In 1741 Christian Goldbach wrote to Leonhard Euler that he was sure “Every number greater than 5 can be written as the sum of 3 primes” but he could not prove it.
3.
(i) If you look at the even numbers on your list and their prime sums you will notice one of the primes in the sum is always a 2. Euler pointed out that this means that every even number greater than 2 can be written at the sum of just 2 primes. Explain why this is true?
(ii) What about the odd numbers? They seem to have a 3 in every prime sum. What if you take the
3 away from each sum? Does this mean we only need to investigate the even numbers? Explain.
Euler reasoned that Goldbach’s Conjecture could be restated “Every even whole number greater than 2 can be expressed as a sum of two primes” . This is one of the oldest unsolved problems in Mathematics.
4.
Find all the different pairs of primes which add to 98.
5.
Some prime numbers have a special property that 2 X prime – 1 = another prime. These prime numbers are called Sophie Germain (1776-1831) primes after the mathematician who investigated them. For example, 23 is a Sophie Germain prime because 2 X 23 - 1 = 47.
Find all the Sophie Germain primes less than 100.
6.
Mersenne primes are named after Marin Mersenne (1588-1648), a French monk and scholar. An example of a Mersenne prime is 7 because 2
3
2
7
1 127
1 7 . 127 is also a Mersenne prime because
Find the other two Mersenne primes in your list of primes less than 100.
7.
As of 2011 there are 47 known Mersenne primes. Of these the largest is
12,978,189 digits which, if printed out would be the size of a novel.
2
4 3 ,1 1 2 ,6 0 9
1 and it has
How is the search for new primes carried out? Hint: GIMPS stands for Great Internet Mersenne
Prime Search.
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