Numbers in their Prime!
Oblong NumbersCan 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 CompositesWe 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, ...
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...
eg: 4 = 2x2; 6 = 2x3; 8 = 2x2x2; 9 = 3x3; 10 = 2x5; 12 = 2x2x3;
See also more on record-breaking prime numbers.
Primes, Factors and Divisibility