Saturday, July 4, 2015

Prime Numbers Are Used For Extraterrestrial Communication

Yes, prime numbers are being used to send messages to aliens.

The Arecibo Message was broadcast into space via radio waves at a ceremony to mark the remodeling of the Arecibo radio telescope in Puerto Rico in 1974. The message was aimed at the current location of globular cluster M13 some 25,000 light years away.

The message consisted of 1,679 binary digits (0's and 1's). The number 1,679 was chosen because it is the product of two primes, 73 and 23. It was felt that any alien getting the message certainly wouldn't know English and maybe wouldn't know how to write. If an alien got the radio message, it would certainly know mathematics, so it would figure out to arrange it as 73 rows of 23 columns by using primes. The message is on the right after it has be so arranged. (Color has been added).

If you wish to see what the message says, go here.

Prime Numbers Are Used To Buy Things

Whenever someone uses a credit card on the internet, prime numbers spring into action.

Before the card number is sent over the internet, it must be encrypted (put into code) for security and once the code is received by the store, it must be decrypted (decoded).

The code that is used the most is called RSA and it is based on prime numbers. It uses a "public key", information that is available to anyone and a "private key" information that only the store has.

The "public key" is a large number that is the product of two large primes and the "private key" is the two large primes themselves.

It is very difficult to factor a given large number into primes, which is what you would have to do to break the code. For example, it took researchers two years to factor a 232-digit number, even using hundreds of parallel computers.

So when a credit card is used, only the "public key" is sent from the store, the message is coded using the key, and sent to the store. The store already has the answer to the key and can easily uncode the message.

Prime Numbers to 400

Ok, if you've read the first blog on prime numbers, you know some rules that help you reduce the number of numbers you have to check to see if a number is prime.

One more rule that may help is the rule that:

All prime numbers are either +1 or -1 from a multiple of 6. Not all +1's and -1's are primes, but all primes are +1 or -1.

What this rule does is move you one away from all the numbers that can be divided by 2 or 3.

Here's another chart of what numbers you have to check as divisors for all primes up to 400.

This chart doesn't prove it, but it gives you hint of a basic rule of all numbers:

Every number is either prime or can be factored (divided) into primes.

This is true just from the way we define prime numbers

Friday, July 3, 2015

Prime Numbers

OK, here's the definition of a prime number:

A prime number (or a prime) is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number.

In other words, a prime number is a counting number that can't be divided by another number.

So what does this mean?

1. First of all, except for 2, a prime number cannot be an even number. If it was even, 2 would divide into it.
2. Secondly, except for 5, any number that ends in a 5 is not prime. If it ends in 5, 5 would divide into it.
3. Thirdly, if the sum of the digits of a number is 3, 6, or 9, the number is not prime. If the sum of the digits is 3, 6, or 9, 3 divides into it.

So it isn't necessary to check for all the possible divisors to tell if a number is prime or not, just the primes less than the number and up to the square root of the number you're checking. (don't worry about the square root, you'll meet it in a year or two.) So this is the way it works for primes up to 100:

So we can now write down all the prime numbers up to 100. Write down your list and compare it to the answer below:

Answer: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 93, 97
Did you get caught with 91? 7 X 13 = 91

Monday, June 29, 2015

I Wonder Why - 34

Take the numbers 1 through 16 and write them into a 4 x 4 grid like so:

 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Now choose a random number in the grid and circle it. Mark out the rest of the numbers in the same row and column.
Do this three times total and you should have one number left. Circle it.

Sum the four numbers you have circled. Why does the sum always equal 34?

Answer: By choosing this way we are picking 4 numbers that do not share rows or columns. The sum of the first column is 1+5+9+13 = 28, but by forcing picks 1, 2 and 3 columns to the right for each row the sum is forced to be 28+1+2+3 = 34.