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

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