Count On


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:

o o o o o o o o

However, that is not the only rectangle. Another is 2 rows of 4 dots:

o o o o
o o o o

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, ...

There are 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: 26972593-1. 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 38th Known Mersenne Prime by George Woltman, pages 367-370 (which you may have to find in the library of your local university).

They 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;
If 1 was included as a prime number, then we can include as many "1" factors as we wanted and there would be many ways to write each number as a product of primes. It is mainly for this reason that 1 is not called a prime number.

See also more on record-breaking prime numbers.

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Primes, Factors and Divisibility