2 Prime NumbersPrime numbers are numbers which have no factors other than 1 and itself. The ancientChinese discovered the primes, but didn’t really do anything with them. It was only inthe Golden Age of the Greeks that the mysteries behind prime numbers wereinvestigated. 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 tonothing? In other words, are there an infinite number of prime numbers?
3 Euclid’s ProofLet’s say there is a finite amount of prime numbers… there must be alargest prime number. Call that number P. That makes our series ofprime 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
4 which numbers are factors? Euclid’s ProofN 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 ofN + 1. That means that N + 1 is prime, since the only other factor can be1. This means there is an infinite number of prime numbers.The largest prime number to have been calculated has 9.1 million digits!
5 Finding PrimesBoth Fermat and Mersenne have offered us some techniques for findingthe prime numbers. Neither always bring up a prime number, but in thecase of Mersenne, his formula leads to finding perfect numbers.Fermat’s formula: 2n + 1Mersenne’s formula: 2n – 1We don’t know whether there are an infinite number of Mersenne primesor whether we can achieve an infinite number of primes from Fermat’sFormula.
6 Density of Primes ≈ 1/log n How Many Primes?Gauss noticed, as Euclid did, that the prime numbers begin to dwindleout 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.)
7 Uses of Prime Numbers There are a number of uses of prime numbers. Some uses have been invented by humans forvarious reasons and some are so engrained innature that it seems prime numbers play a morefundamental role in life than we sometimesrealise…
8 for prime numbers in his music: Music of the PrimesOlivier Messiaen, a famous composer found a great usefor prime numbers in his music:Messiaen used both a 17 and 29 sequence in his piece of musicQuartet for the End of Time. Both motifs start at the same time, however,since they are both prime numbers, the same sequence of notes playingtogether from each sequence wont be the same until they have playedthrough 17 x 29 times each. He held prime numbers very close to hisheart and believed they gave his music a timelessness quality.
9 Primes In Nature Similarly, the cicada, a burrowing insect owes its survival to prime numbers and theirproperties. The cicada lives underground for 17years, making no sound or showing any signs forthis amount of time. After 17 years, all of the insectsappear in the forest for just six weeks to mate beforedying out.
10 Primes In NatureThis survival technique, whilst the noise drives local residentsto evacuate the area for the six weeks due to the noise, hascome about due to a predator that would appear in the forestat regular intervals. The cicada could avoid confrontation withthe predator more often by only appearing every 17 years asthe number is prime. As with Messiaen’s music, it would be along time before the predator and the cicada would meetagain.
11 Prime Numbers In Code Breaking Prime numbers assist us more in today’s society thansome people realise. Internet banking, shopping andgeneral interaction would not be secure if it wasn’t forthese interesting numbers. A particular feature ofprime factors comes in very useful in keeping detailsprivate…
12 Prime Numbers In Code Breaking Codes used to be kept entirely private. Theencoded message, the key to decoding themessage… everything was confidential. However,today there exist a technique that allows encodedmessages and even the method to unlocking themessage to be publicly announced.
13 Prime Numbers In Code Breaking To encode a message…If we want to send the message “HELLO” we simply convert it into astring of numbers:(A=01, B=02… etc.)We can then raise that number to a publicly announced power, divide itby another number which has again been publicly announced and wewill be left with a remainder. This is our encoded message…
14 Prime Numbers In Code Breaking To decode this message…The person who received the coded string of numberswould raise that number to another power whichwould only be known to them. They then divide itagain by the number publicly announced earlier andthe remainder from that would be the string ofnumbers that break down to say “HELLO”!
15 Prime Numbers In Code Breaking For Example…Let our message be “E”. “E” is converted to 05, and is thenraised to the 7th power (this is important and will be explainedlater). Our number is now 78,125. We divide that number by33 (again will be explained later) to give 2367 with aremainder of 14.14 is our encoded message.
16 Prime Numbers In Code Breaking Now, to decode…We raise 14 to the 3rd power to give We dividethat number by 33 which gives 83 with a remainder of5…5 is our decoded message and converts to “E”, theoriginal message.
17 Prime Numbers In Code Breaking 33 is the key in this code breaking scenario. There is amathematical occurrence deep within this numberconcerning its prime factors, 3 and 11.If we multiply the numbers that are one less than thefactors and add one we get another number. So;(3-1) x (11-1) + 1 = 21
18 Prime Numbers In Code Breaking We can then split the resulting number (21) into itsprime factors, 3 and 7.Notice that these are the powers used to code anddecode the message. This procedure caries throughwith all numbers, no matter how big they are. This isprecisely why the coding works.
19 Prime Numbers In Code Breaking Now, splitting 33 into its prime factors isn’t really that difficult. However, imagine you were given:…
20 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, itsimply takes far too much time to decode themessages.
21 Prime Numbers In Code Breaking The numbers used as coding and decoding powers dependentirely on the technology available at the time and theamount 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 cansometimes struggle to cope with.
22 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!