1. P RIME N UMBERS

2. W HAT IS A P RIME N UMBER ? In mathematics, a prime number (or a prime) is a natural number which has exactly two distinct natural number divisors: 1 and itself. The first twenty-five prime numbers are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97 1 is NOT a prime number. WHY?...

3. ... BECAUSE * Until the 19th century, number 1 was considered a prime, with the definition being just that a prime is divisible only by 1 and itself but not requiring a specific number of distinct divisors. The change occurred so that the fundamental theorem of arithmetic, as stated, is valid, i.e., “each number has a unique factorisation into primes.” The number one is useless in this regard because a = 1.a = 1.1.a =... That is, divisibility by one fails to provide us any information about a.

4. E UCLID ’ S P RIMES Euclid first proved that the number of primes is infinite. There is no largest prime number as much as there is no largest number! Euclid started by looking at the known primes and adding one to their product. For example both 2 and 3 are primes: their product + 1 is also a prime: 2*3+1=7. Continue by adding 5 to the same process: 2*3*5+1=31 – and yes that is a prime too.

5. E UCLID ’ S P ROOF oCoCall the primes in our finite list p 1, p 2,..., p r. oLoLet P be any common multiple of these primes plus one (for example, P = p 1 p 2...p r +1). oNoNow P is either prime or it is not. If it is prime, then P is a prime that was not in our list. If P is not prime, then it is divisible by some prime, call it p. oNoNotice p can not be any of p 1, p 2,..., p r, otherwise p would divide 1, which is impossible. oSoSo this prime p is some prime that was not in our original list. Either way, the original list was incomplete.

6. * oNoNote that what is found in this proof is another prime - one not in the given initial set. oToThere is no size restriction on this new prime, it may even be smaller than some of those in the initial set. oFoFor example, if we begin with the set: {2, 3, 7, 43, 13, 139, 3263443} then P = 547. 607. 1033. 31051. oToThe new prime found would be 547, 607, 1033 or 31051, all of which are smaller than the last prime in the original set.

7. C AN N EGATIVE N UMBERS B E P RIME ? By the usual definition of prime for integers, negative integers can not be prime.prime By this definition, primes are integers greater than one with no positive divisors besides one and itself. Negative numbers are excluded. In fact, they are given no thought.

8. B UT, WHEN VIEWED DIFFERENTLY … Now suppose we want to bring in the negative numbers: then -a divides b when every a does, so we treat them as essentially the same divisor. This happens because -1 divides 1, which in turn divides everything. Numbers that divide one are called units. Two numbers a and b for which a is a unit times b are called associates. So the divisors a and -a of b above are associates. In the same way, -3 and 3 are associates, and in a sense represent the same prime.

9. So yes, negative integers can be prime (when viewed this way). In fact the integer -p is prime whenever p, but since they are associates, we really do not have any new primes. For example: the Gaussian integers are the complex numbers a+bi where a and b are both integers. There are four units (integers that divide one) in this number system: 1, -1, i, and -i. So each prime has four associates.

10. U SES OF P RIMES Calculation of prime numbers was a purely mathematical endeavour. But, in the 19th century, there was a need for secrecy, especially during times of war. Messages and files needed to be e n c o d ℮ d, so that the enemy couldn’t read them. Encryption was used, and computers were used to make more complex, harder to crack codes. It was found that using two prime numbers multiplied together makes a much better key than any old number, because it has only four factors. One, itself, and the two primes that it is a product of. This makes the code much harder to crack.

11. I NSECTS Primes only use is in the computer world, and we still don’t have any use for primes in the physical world, however some say that some insects do. Some insects will live in the ground for a number of years, and come out after 13 or 17 years. Both 13 and 17 are prime numbers, and by emerging at these times, it makes it harder for predators to adapt and kill the insects, and therefore more of them survive. One bug that does, this is the cicada.

12. EARLY PRIMES Writers thought that the numbers of the form 2 n -1 were prime for all primes n, but in 1536 Hudalricus Regius showed that 2 11 -1 = 2047 was not prime (it is 23 * 89). By 1603 Pietro Cataldi correctly verified that 2 17 -1 and 2 19 -1 were both prime, but incorrectly stated 2 n -1 was also prime for 23, 29, 31 and 37. In 1640 Fermat showed Cataldi was wrong about 23 and 37. In 1738 Euler showed Cataldi was also wrong about 29. However, Euler also showed Cataldi's statement about 31 was correct.

13. MERSENNE PRIME A French Monk, Marin Mersenne stated that the numbers 2 22 2n -1 were prime for n = 2, 3, 5, 7, 13, 17, 19, 31, 67, 127 and 257 Mersenne's assumption was incorrect but still got his name attached to these numbers. When 2 n -1 is prime it is said to be a Mersenne prime. It was not until over 100 years later, in 1750, that Euler verified the next number on Mersenne's and Regius' lists, 2 31 -1, was prime.

14. ... A century later in 1876, Lucas verified 2 127 -1 was also prime. Seven years later Pervouchine showed 2 61 -1 was prime, which Mersenne missed. In the early 1900's Powers (computer software) showed that Mersenne also missed the primes 2 89 -1 and 2 107 - 1. Finally, by 1947 Mersenne's range, n < 258, had been completely checked and it was determined that the correct list is: n = 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107 and 127.

15. LARGEST KNOWN PRIME The largest known prime verified before computers was in 1951. Ferrier found the prime (2 148 +1)/17= 20988936657440586486151264256610222593863921 The largest known prime now is a 12,978,189 digit Mersenne prime 2 43112609 -1 which was found in August 2008.