# Count On

## Explorer

#### Numbers in their Prime!

##### Oblong Numbers

Can we arrange a given number of dots into a rectangle?Take, for instance, 8. We can arrange it into a simple flat rectangle which is just all 8 dots in a row:

There aren't any more shapes, only the same rectangles turned round to make 8 rows of 1 dot and 4 rows of 2 dots.

##### Is Every number Oblong?

Can we find a rectangle for any number of dots?Yes, we can, since a single row of the dots will do and this is possible for any length of row.

The more interesting question is whether this is the

*only*solution or not.

We will call a number **oblong** if it has a "genuine" rectangular
array of dots as well as the simple single row solution.

The question now is,

*Is every number Oblong?*

We have seen that 8 is. 9 is also since it is 3x3 as is 6=2x3 and 4=2x2, but 5 is not oblong, and neither is 7, or 3 or 2.

##### Primes and Composites

We can now just look at writing a number N as a product of two numbers - called**factors**- and see if it has any other solution apart from 1xN.

The process of discovering such products is called **factorisation** and
oblong numbers are also called **composite numbers** and those with only the
simple shape (one row) are called **prime numbers**.

Numbers whose only factors are 1 and themselves are called prime numbers; Numbers with more than these two factors are called composite numbers. So every whole number is either prime or else it is composite. |

E.g. 8 is composite (8=2x4), 11 is prime.

The list of all the prime numbers begins: 2, 3, 5, 7, 11, 13, 17, 19, ... |

*infinitely many prime numbers*, so our list of prime numbers will continue to grow without ever coming to an end. On June 1, 1999, the latest

*largest known prime number*was found - a number so large that it has 2,098,960 digits: 2

^{6972593}-1. It took Nayan Hajratwala 111 days running part-time on a 350 MHz Pentium II computer and the result was then verified by three different computers using different software.

You can read more about it in

*The Fibonacci Quarterly*, volume 37, November 1999, in an article entitled

**On the Discovery of the 38**by George Woltman, pages 367-370 (which you may have to find in the library of your local university).

^{th}Known Mersenne PrimeThey are called **prime numbers** because...

*...every other number can be made from multiples of prime numbers in essentially one way only:*

eg: 4 = 2x2; 6 = 2x3; 8 = 2x2x2; 9 = 3x3; 10 = 2x5; 12 = 2x2x3;

See also more on *record-breaking prime numbers*.

back toPrimes, Factors and Divisibility |