prime numbers by becky james. prime numbers prime numbers are numbers which have no factors other...

Prime NumbersPrime Numbers

By Becky JamesBy Becky James

Prime NumbersPrime NumbersPrime numbers are numbers which have no factors other than 1 and itself. The ancientPrime 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 inChinese 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 were the Golden Age of the Greeks that the mysteries behind prime numbers were investigated. Euclid in his investigated. Euclid in his ElementsElements offered some insight into prime numbers: offered some insight into prime numbers:

If you look at the prime numbers between 1and 20…If you look at the prime numbers between 1and 20…

2, 3, 5, 7, 11, 13, 17, 19.2, 3, 5, 7, 11, 13, 17, 19.

There are 8, nearly half are prime. Between 20 and 40…There are 8, nearly half are prime. Between 20 and 40…

23, 29, 31, 37.23, 29, 31, 37.

There are 4. The number of primes is reducing, so do they eventually dwindle out to 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?nothing? In other words, are there an infinite number of prime numbers?

Euclid’s ProofEuclid’s Proof

Let’s say there is a finite amount of prime numbers… there must be aLet’s say there is a finite amount of prime numbers… there must be alargest prime number. Call that number P. That makes our series oflargest prime number. Call that number P. That makes our series ofprime numbers look like this:prime numbers look like this:

2, 3, 5, 7……. P.2, 3, 5, 7……. P.

If we multiply all of those numbers together, we get….If we multiply all of those numbers together, we get….

2 x 3 x 5 x 7 x……. x P = N2 x 3 x 5 x 7 x……. x P = N

Euclid’s ProofEuclid’s Proof

N is the product of all the primes.N is the product of all the primes.

If we now consider N + 1…If we now consider N + 1…

which numbers are factors?which numbers are factors?

None! 2 is the smallest factor of N, so therefore cannot be a factor of 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 beN + 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. 1. This means there is an infinite number of prime numbers.

The largest prime number to have been calculated has 9.1 million digits!The largest prime number to have been calculated has 9.1 million digits!

Finding PrimesFinding Primes

Both Fermat and Mersenne have offered us some techniques for finding Both Fermat and Mersenne have offered us some techniques for finding the prime numbers. Neither always bring up a prime number, but in thethe prime numbers. Neither always bring up a prime number, but in thecase of Mersenne, his formula leads to finding perfect numbers.case of Mersenne, his formula leads to finding perfect numbers.

Fermat’s formula: 2Fermat’s formula: 2nn + 1 + 1

Mersenne’s formula: 2Mersenne’s formula: 2nn – 1 – 1

We don’t know whether there are an infinite number of Mersenne primes 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 or whether we can achieve an infinite number of primes from Fermat’s Formula.Formula.

How Many Primes?How Many Primes?

Gauss noticed, as Euclid did, that the prime numbers begin to dwindle Gauss noticed, as Euclid did, that the prime numbers begin to dwindle out as we get higher and higher up the number ladder…out as we get higher and higher up the number ladder…

After lots of calculating and trial and error, Gauss showed that:After lots of calculating and trial and error, Gauss showed that:

Density of Primes ≈ 1/log n Density of Primes ≈ 1/log n

(where n = the sample of numbers in which the primes are being counted.)(where n = the sample of numbers in which the primes are being counted.)

Uses of Prime NumbersUses of Prime Numbers

There are a number of uses of prime numbers. There are a number of uses of prime numbers. Some uses have been invented by humans for Some uses have been invented by humans for various reasons and some are so engrained in various reasons and some are so engrained in nature that it seems prime numbers play a more nature that it seems prime numbers play a more

fundamental role in life than we sometimes fundamental role in life than we sometimes realise…realise…

Music of the PrimesMusic of the Primes

Olivier Messiaen, a famous composer found a great use Olivier Messiaen, a famous composer found a great use for prime numbers in his music: for prime numbers in his music:

Messiaen used both a 17 and 29 sequence in his piece of music Messiaen used both a 17 and 29 sequence in his piece of music Quartet for the End of TimeQuartet for the End of Time. Both motifs start at the same . Both motifs start at the same

time, however, time, however, since they are both prime numbers, the same sequence of notes playing since they are both prime numbers, the same sequence of notes playing together from each sequence wont be the same until they have played 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 through 17 x 29 times each. He held prime numbers very close to his heart and believed they gave his music a timelessness quality.heart and believed they gave his music a timelessness quality.

Primes In NaturePrimes In Nature

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

Primes In NaturePrimes In Nature

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

long time before the predator and the cicada would meet long time before the predator and the cicada would meet again.again.

Prime Numbers In Code Prime Numbers In Code BreakingBreaking

Prime numbers assist us more in today’s society than Prime numbers assist us more in today’s society than some people realise. Internet banking, shopping and some people realise. Internet banking, shopping and general interaction would not be secure if it wasn’t for general interaction would not be secure if it wasn’t for

these interesting numbers. A particular feature of these interesting numbers. A particular feature of prime factors comes in very useful in keeping details prime factors comes in very useful in keeping details

private…private…


Codes used to be kept entirely private. The Codes used to be kept entirely private. The encoded message, the key to decoding the encoded message, the key to decoding the

message… everything was confidential. However, message… everything was confidential. However, today there exist a technique that allows encoded today there exist a technique that allows encoded messages and even the method to unlocking the messages and even the method to unlocking the

message to be publicly announced.message to be publicly announced.


To encode a message…To encode a message…

If we want to send the message “HELLO” we simply convert it into a If we want to send the message “HELLO” we simply convert it into a string of numbers:string of numbers:

08051212160805121216(A=01, B=02… etc.)(A=01, B=02… etc.)

We can then raise that number to a publicly announced power, divide it We can then raise that number to a publicly announced power, divide it by another number which has again been publicly announced and we by another number which has again been publicly announced and we will be left with a remainder. This is our encoded message…will be left with a remainder. This is our encoded message…


To decode this message…To decode this message…

The person who received the coded string of numbers The person who received the coded string of numbers would raise that number to another power which would raise that number to another power which would only be known to them. They then divide it would only be known to them. They then divide it

again by the number publicly announced earlier and again by the number publicly announced earlier and the remainder from that would be the string of the remainder from that would be the string of

numbers that break down to say “HELLO”!numbers that break down to say “HELLO”!


For Example…For Example…

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

14 is our encoded message. 14 is our encoded message.


Now, to decode…Now, to decode…

We raise 14 to the 3We raise 14 to the 3rdrd power to give 2744. We divide power to give 2744. We divide that number by 33 which gives 83 with a remainder of that number by 33 which gives 83 with a remainder of 5…5…

5 is our decoded message and converts to “E”, the 5 is our decoded message and converts to “E”, the original message.original message.


33 is the key in this code breaking scenario. There is a 33 is the key in this code breaking scenario. There is a mathematical occurrence deep within this number mathematical occurrence deep within this number

concerning its prime factors, 3 and 11. concerning its prime factors, 3 and 11.

If we multiply the numbers that are one less than the If we multiply the numbers that are one less than the factors and add one we get another number. So;factors and add one we get another number. So;

(3-1) x (11-1) + 1 = 21(3-1) x (11-1) + 1 = 21


We can then split the resulting number (21) into its We can then split the resulting number (21) into its prime factors, 3 and 7.prime factors, 3 and 7.

Notice that these are the powers used to code and Notice that these are the powers used to code and decode the message. This procedure caries through decode the message. This procedure caries through with all numbers, no matter how big they are. This is with all numbers, no matter how big they are. This is precisely why the coding works.precisely why the coding works.


Now, splitting 33 into its prime factors isn’t Now, splitting 33 into its prime factors isn’t really that difficult. However, imagine you really that difficult. However, imagine you were given:were given:

3445763848233475648765873462386476547678947568436584344576384823347564876587346238647654767894756843658475687568

3682376486435236423842873468273638764283648236435736368237648643523642384287346827363876428364823643573643294329

8472903746436486348364877455476875764536507865544537847290374643648634836487745547687576453650786554453765456545

4358438573458739579047593472398479857435676593274029435843857345873957904759347239847985743567659327402938743874

9479487683746293479238794563475623846902374902347…9479487683746293479238794563475623846902374902347…


Splitting that number, with hundreds of digits, into two Splitting that number, with hundreds of digits, into two prime factors would take even the fastest computer prime factors would take even the fastest computer in the world more time to crack it than the Universe in the world more time to crack it than the Universe

has existed. has existed.

Unless by a fluke the prime factors are found, it Unless by a fluke the prime factors are found, it simply takes far too much time to decode the simply takes far too much time to decode the

messages.messages.


The numbers used as coding and decoding powers depend The numbers used as coding and decoding powers depend entirely on the technology available at the time and the entirely on the technology available at the time and the amount of time it would take a computer to factor a number.amount of time it would take a computer to factor a number.

Since the messages tend to be a lot longer than “E” or Since the messages tend to be a lot longer than “E” or ““HELLO” the process becomes longer and more complicated, HELLO” the process becomes longer and more complicated, which unfortunately the finite nature of technology can which unfortunately the finite nature of technology can sometimes struggle to cope with.sometimes struggle to cope with.


However, since no-ones knows of a way of However, since no-ones knows of a way of quickly factorising a number into prime factors quickly factorising a number into prime factors

the process is quite safe for now!the process is quite safe for now!

prime numbers by becky james. prime numbers prime numbers are numbers which have no factors other...

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