12. Fermat’s Last Theorem (and Other Conjectures)


On 27 June 1997 Andrew Wiles collected the Wolfskehl prize which because of World War inflation was then worth only €40,000. He said, ‘having solved this problem there is a sense of loss, but at the same time there is a sense of freedom. I was so obsessed by this problem that for eight years I was thinking about it all the time, when I woke in the morning to when I went to sleep at night. That is a long time to think of one thing. That particular odyssey is now over. My mind is at rest.’


Prime Numbers


A prime number is a natural number which has no proper factors. A prime number has two divisors, itself and 1. Therefore, 1 is not a prime number.


Look at all the numbers between 1 and 20 inclusive: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20


Remove 1 as it is not prime. Take out the numbers which are divisible by 2 (excluding 2 itself). This leaves: 2, 3, 5, 7, 9, 11, 13, 15, 17, 19


Take out the numbers which 3 divides into (excluding 3 itself), this leaves:


2, 3, 5, 7, 11, 13, 17, 19


Pierre de Fermat: French amateur mathematician, (1601–1665) Known as


The


Riddler, Fermat formed many theorems. Although he devoted much time to mathematics, he was not a professional mathematician. Instead, he preferred to send letters to his friends with his most recent theorem but he did not send any accompanying proof. In this way, he managed to receive fame as a mathematician without having to prove his theorems. Along with Pascal, he developed the basis for the theory of probability. One mystery which still troubles some mathematicians is ‘Did Fermat have a simple proof for his famous last theorem as he claimed?’ Most mathematicians doubt it but there are some who hope to find fame and glory by finding the original proof.


The numbers 4 divides into are already gone (it divides into 2). Then check 5 and so on. The full list of primes between 1 and 20 inclusive is:


2, 3, 5, 7, 11, 13, 17, 19 PROJECT 12.2 Prime Numbers


List all the prime numbers between 2 and 100 inclusive. There are 25 of them.


There is a lot of interest in prime numbers, especially to cryptographers. Codes based on multiplying large prime numbers together are very difficult to break. To some extent the success of computer-based financial transactions depends on the security provided by such codes.


PROJECT 12.3 Credit Card Security


What property of prime numbers makes them so important for ensuring the security of credit card numbers during online transactions? Working in small groups of 2-4, carry out some research on the internet and write a note on this topic (search using terms like ‘credit card’, ‘security’, and ‘prime numbers’).


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