Download What are the Prime Numbers?

WHAT ARE THE PRIME
NUMBERS?
PRIME NUMBERS:
• Are “building blocks” for the integers:
1,2,3,4,5,6,7,8,…
• Integers with only two factors (1 and itself)
Examples:
2,3,5,7,11,13,17,19,23,…
COMPOSITE NUMBERS
• Are integers that are not prime
Examples:
4 = 2*2 (has factors 1,2, and 4)
6 = 2*3
14 = 2*7
2007 = 3*3*223
Determine whether the given
number is prime or composite.
1)
2)
3)
4)
5)
24
28
5
25
33
CHALLENGE:
If x is any integer,
then
Can 2x-1 ever be a
prime number?
Explain why?
Determine whether the given
number is prime or composite.
1)
2)
3)
4)
5)
24 = 2*2*2*3
28 = 2*2*7
5 is prime
25 = 5*5
33 = 3*11
CHALLENGE:
If x is any integer, then
Can 2x-1 ever be a prime number?
YES!
x=2 since 22-1 = 3
MERSENNE PRIMES
• Primes numbers
of the form
2x-1 are called
Mersenne
primes.
x
2x-1
2
3
3
7
5
31
7
127
12
8191
Largest known prime number
32,582,657
2
-1
• It has 9,808,358 digits!
• $100,000 prize for first person
to find 10 million digit prime
number.
How big is that?
• Compare that to the number
of suspected atoms in the
universe, a number with
around only 80 digits.
IMPORTANCE OF PRIME
NUMBERS
• Encoding and security of document
transfers via the internet or other sources
• Crucial and ubiquitous to the subject of
number theory.
Your mom at the internet
• Purchasing items,
email accounts,
bank transactions,
and more
Are made secure
with the help of
prime numbers and
mathematicians!
How do they make it secure?
• Let x and y be “large” prime numbers.
x*y = LARGE #
Mathematicians proved that the size of your
prime numbers x and y determines the
length of time to break the code.
Try this:
7 * 32 * 5 * 7
=
2
• Factor 40320
Wait just a minute…
Mathematicians proved that there are
infinitely many primes numbers,
and they determined ways to find “really
large” ones.
Multiplying two prime numbers x and y
(each having say 201 digits)
gives a 402 digit number
x*y = LARGE #.
It is known that there are about 10400
prime numbers less than our number
LARGE #.
• In order to factor our number
LARGE #,
It will take WAY TOO LONG to
look through 10400 prime
numbers!
"My money is safe!”
WHAT ARE THE PRIME
NUMBERS?
Prepared by Mr. Baczkowski
for Blythewood Middle School
th
8 grade science class
“We love science class!!!!!!”