Transcript Prime Numbers - Oldham Sixth Form College

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
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.
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…
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!