4] and .
In some cases, where the proof is short and useful, it will be presented.
However, there are some results too, the proof of which is extremely difficult.
In these cases text-books usually refer to an original source.
Through our discussion we present the short history of this field, with the
names of the important authors. More historical results can be found, besides
the text-books e.g. in ,  and .
The illustration of theoretical results contains Maple examples and short
programs. These parts suppose, that the reader has some Maple language
expertise. Thorough description of the use of Maple can be found e.g. in
,  and .
Prime numbers and irreducible numbers We consider in the following integer numbers, i.e. numbers in . If a number is a divisor of , we denote it by .
DEFINITION 1. Prime property For a number let . If from this or follows, then we call the number a prime number.
DEFINITION 2. Irreducible property We call a number irreducible, if from follows or .
THEOREM 1. A number is prime if and only if it is irreducible.
REMARKS 1. This theorem usually does not hold if we leave the ring . It is true in every case, that a prime element is irreducible. But the other direction gets violated in many rings. For example in the number is irreducible, but does not imply, that or . 2. Not prime elements are called composite numbers.
THEOREM 2. The base theorem of number theory Every number differ from 0, -1 and 1 can be written as the product of finite irreducible numbers. This factorization is unique regardless of the numbers 1, -1 and the rank of the factors.
In the Maple system we can factorize a number with the function