Prime integers play a very important role in our lives. Even though you
may have not known it, we see examples of prime integers every day of our lives.
Prime numbers are used in such things as cryptology. Every time you use a
credit card prime numbers come into play. Each business chooses two large primes,
x and y, which they keep secret. The product of these primes,
Z=x*y, is made public. A calculation using Z encrypts your credit card,
but the only way to undo the calculation and decrypt the secret message is to
know the secret primes x and y.
Prime numbers also appear in nature. The insect know as the
"periodical cicada" has a life-cycle that revolves around a prime number.
The "periodical cicada" is born underground and after 17 years, they come
to the surface to mate, lay their eggs, and die. It is believed that this
is because their life-cycle is like this in order to avoid other parasites
of a life cycle that would be divided by theirs. Since 17 is somewhat
large for a life-cycle and it is not divisible by any shorter life-cycles,
the cicadas will rarely meet its parasite.
A prime integer is an integer that is divided by exactly two different
positive integers, 1 and itself.
| A positive integer p greater than 1 is called prime if the only positive factors of p are 1 and p. A positive integer that is greater than 1 and is not prime is called composite. |
Therefore an example of a prime integer is 11 because its only positive factors are 1 and 11. On the other hand, 6 is not a prime number because it is divisible by 1,2,3, and 6.
| Every positive integer greater than 1 can be written uniquely as a prime or as the product of two or more primes where the prime factors are written in order of nondecreasing size. |
This theorem introduces an interesting way to look at every integer. It shows how important prime numbers are to all integers.
Examples: The prime factorizations for 25, 322, and 1136 are:| If n is a composite integer, then n has a prime divisor less than or equal to the √n |
To show an example of this, we will use 143
√ 143=approximately 11.96, so we have to use integers less than 12.
The primes not exceeding √ 143 are 2,3,5,7, and 11. Since
11*13=143, 143 is divisible by 11 which is prime and less than
√ 143.
There are an infinite number of primes. This was proven by Euclid, a great mathematician.
There are constantly larger and larger prime numbers being found. Two years ago a record prime number was found that has over 4 million digits.
Mersenne primes are known as primes of the form 2p-1, where p is also prime.
However this does not hold true for every p.
An example of a Mersenne prime is 25-1 = 31.
An example of a non-Mersenne prime is 211-1 = 2047, but 2047 is
no prime. 23*89=2047.
The largest Mersenne prime known so far is 213,466,917-1.
| The ratio of the number of primes not exceeding x and x/ln x approaches 1 as x grows without bound.(Here lnx is the natural logarithm of x.) |
This theorem can help us estimate the odds that a randomly chosen number of a certain size is prime. For example the odds that an integer near 100 is prime are approximately 1/ln 100, which is approximately 217/1000.
As you can see, prime numbers have many interesting properties and applications. Before we move on to greatest common divisors and least common multiples, here are a few quick questions.
1. Are these integers prime?
a.27
b.37
c.111
d.133
2. Find the prime factorization of 76 and
147.