Learning Session 4: Prime Number Investigation Prime numbers ...

Learning Session 4: Prime Number InvestigationPrime numbers have fascinated mathematicians from many cultures for thousands of years. Nowadays theyare important because the best commercial and military ciphers depend on their properties. In fact largeprime 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 primenumbers to 100. Remember that a prime number can be divided only by itself and one and 2 is thesmallest prime.1 2 3 4 5 6 7 8 9 1011 12 13 14 15 16 17 18 19 2021 22 23 24 25 26 27 28 29 3031 32 33 34 35 36 37 38 39 4041 42 43 44 45 46 47 48 49 5051 52 53 54 55 56 57 58 59 6061 62 63 64 65 66 67 68 69 7071 72 73 74 75 76 77 78 79 8081 82 83 84 85 86 87 88 89 9091 92 93 94 95 96 97 98 99 100The 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 crossedare not primes, because they are multiples of other numbers. The numbers that arecircled 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+27 = 2+2+38 = 2+3+3Record your findings and anything you think might be of interest. Do you want to change youranswer to the question?Page 1

In 1741 Christian Goldbach wrote to Leonhard Euler that he was sure “Every number greater than 5 can bewritten 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 theprimes in the sum is always a 2. Euler pointed out that this means that every even number greaterthan 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 the3 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 2can 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 primenumbers are called Sophie Germain (1776-1831) primes after the mathematician who investigatedthem. 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. Anexample of a Mersenne prime is 7 because 2 3 1 7 . 127 is also a Mersenne prime because2 7 1 127Find 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 2 43,112,609 1and it has12,978,189 digits which, if printed out would be the size of a novel.How is the search for new primes carried out? Hint: GIMPS stands for Great Internet MersennePrime Search.Page 2