Prime Numbers By Becky James

Prime Numbers Prime numbers are numbers which have no factors other than 1 and itself. The ancient Chinese discovered the primes, but didn’t really do anything with them. It was only in the Golden Age of the Greeks that the mysteries behind prime numbers were investigated. Euclid in his Elements offered some insight into prime numbers: If you look at the prime numbers between 1and 20… 2, 3, 5, 7, 11, 13, 17, 19. There are 8, nearly half are prime. Between 20 and 40… 23, 29, 31, 37. There are 4. The number of primes is reducing, so do they eventually dwindle out to nothing? In other words, are there an infinite number of prime numbers?

Euclid’s Proof Let’s say there is a finite amount of prime numbers… there must be a largest prime number. Call that number P. That makes our series of prime numbers look like this: 2, 3, 5, 7……. P. If we multiply all of those numbers together, we get…. 2 x 3 x 5 x 7 x……. x P = N

which numbers are factors? Euclid’s Proof N is the product of all the primes. If we now consider N + 1… which numbers are factors? None! 2 is the smallest factor of N, so therefore cannot be a factor of N + 1. That means that N + 1 is prime, since the only other factor can be 1. This means there is an infinite number of prime numbers. The largest prime number to have been calculated has 9.1 million digits!

Finding Primes Both Fermat and Mersenne have offered us some techniques for finding the prime numbers. Neither always bring up a prime number, but in the case of Mersenne, his formula leads to finding perfect numbers. Fermat’s formula: 2n + 1 Mersenne’s formula: 2n – 1 We don’t know whether there are an infinite number of Mersenne primes or whether we can achieve an infinite number of primes from Fermat’s Formula.

Density of Primes ≈ 1/log n How Many Primes? Gauss noticed, as Euclid did, that the prime numbers begin to dwindle out as we get higher and higher up the number ladder… After lots of calculating and trial and error, Gauss showed that: Density of Primes ≈ 1/log n (where n = the sample of numbers in which the primes are being counted.)

Uses of Prime Numbers There are a number of uses of prime numbers. Some uses have been invented by humans for various reasons and some are so engrained in nature that it seems prime numbers play a more fundamental role in life than we sometimes realise…

for prime numbers in his music: Music of the Primes Olivier Messiaen, a famous composer found a great use for prime numbers in his music: Messiaen used both a 17 and 29 sequence in his piece of music Quartet for the End of Time. Both motifs start at the same time, however, since they are both prime numbers, the same sequence of notes playing together from each sequence wont be the same until they have played through 17 x 29 times each. He held prime numbers very close to his heart and believed they gave his music a timelessness quality.

Primes In Nature Similarly, the cicada, a burrowing insect owes its survival to prime numbers and their properties. The cicada lives underground for 17 years, making no sound or showing any signs for this amount of time. After 17 years, all of the insects appear in the forest for just six weeks to mate before dying out.

Primes In Nature This survival technique, whilst the noise drives local residents to evacuate the area for the six weeks due to the noise, has come about due to a predator that would appear in the forest at regular intervals. The cicada could avoid confrontation with the predator more often by only appearing every 17 years as the number is prime. As with Messiaen’s music, it would be a long time before the predator and the cicada would meet again.

Prime Numbers In Code Breaking Prime numbers assist us more in today’s society than some people realise. Internet banking, shopping and general interaction would not be secure if it wasn’t for these interesting numbers. A particular feature of prime factors comes in very useful in keeping details private…

Prime Numbers In Code Breaking Codes used to be kept entirely private. The encoded message, the key to decoding the message… everything was confidential. However, today there exist a technique that allows encoded messages and even the method to unlocking the message to be publicly announced.

Prime Numbers In Code Breaking To encode a message… If we want to send the message “HELLO” we simply convert it into a string of numbers: 0805121216 (A=01, B=02… etc.) We can then raise that number to a publicly announced power, divide it by another number which has again been publicly announced and we will be left with a remainder. This is our encoded message…

Prime Numbers In Code Breaking To decode this message… The person who received the coded string of numbers would raise that number to another power which would only be known to them. They then divide it again by the number publicly announced earlier and the remainder from that would be the string of numbers that break down to say “HELLO”!

Prime Numbers In Code Breaking For Example… Let our message be “E”. “E” is converted to 05, and is then raised to the 7th power (this is important and will be explained later). Our number is now 78,125. We divide that number by 33 (again will be explained later) to give 2367 with a remainder of 14. 14 is our encoded message.

Prime Numbers In Code Breaking Now, to decode… We raise 14 to the 3rd power to give 2744. We divide that number by 33 which gives 83 with a remainder of 5… 5 is our decoded message and converts to “E”, the original message.

Prime Numbers In Code Breaking 33 is the key in this code breaking scenario. There is a mathematical occurrence deep within this number concerning its prime factors, 3 and 11. If we multiply the numbers that are one less than the factors and add one we get another number. So; (3-1) x (11-1) + 1 = 21

Prime Numbers In Code Breaking We can then split the resulting number (21) into its prime factors, 3 and 7. Notice that these are the powers used to code and decode the message. This procedure caries through with all numbers, no matter how big they are. This is precisely why the coding works.

Prime Numbers In Code Breaking Now, splitting 33 into its prime factors isn’t really that difficult. However, imagine you were given: 34457638482334756487658734623864765476789475684365847568 36823764864352364238428734682736387642836482364357364329 84729037464364863483648774554768757645365078655445376545 43584385734587395790475934723984798574356765932740293874 9479487683746293479238794563475623846902374902347…

Prime Numbers In Code Breaking Splitting that number, with hundreds of digits, into two prime factors would take even the fastest computer in the world more time to crack it than the Universe has existed. Unless by a fluke the prime factors are found, it simply takes far too much time to decode the messages.

Prime Numbers In Code Breaking The numbers used as coding and decoding powers depend entirely on the technology available at the time and the amount of time it would take a computer to factor a number. Since the messages tend to be a lot longer than “E” or “HELLO” the process becomes longer and more complicated, which unfortunately the finite nature of technology can sometimes struggle to cope with.

Prime Numbers In Code Breaking However, since no-ones knows of a way of quickly factorising a number into prime factors the process is quite safe for now!