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Factoring And Prime Numbers - A Brief Introduction Part 1

By Patrick Attlee

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Published: 21Aug2009
Word count: 584
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A large part of that branch of mathematics somewhat loosely termed "Number Theory" is either devoted to, or has its origins in the quest for prime numbers. The determination of whether a given large number is prime, or if not what are its prime factors, is frequently a matter of interest and more often than not one of considerable difficulty. In this article we shall examine some of the methods which have been developed for reducing these difficulties.

Large numbers or small, there are of course some elementary observations which can be made at once. Even numbers and multiples of 5 are recognisable at sight, multiples of 3 respond to a simple and rapid mental check and thanks to a lucky combination of factors those of 7, 11 13 and 37 are found with little effort.

Many other of the smaller primes can be eliminated without having to use the laborious process of testing by actual division. These methods obviously dispose of vast quantities of the natural numbers-even numbers and multiples of 3 alone take care of two-thirds of them, for instance-but there remains an infinity of integers which are either prime or have larger prime factors than these methods are equipped to deal with. These can be divided broadly into two classes; in the one case numbers of a particular structure such as, say, those expressed generally by the formula xn +- 1 for which the 'form' of possible factors can usually be determined fairly easily. On the other hand there are amorphous numbers, whose factor forms can only be found by congruence techniques which for numbers of six or seven digits are inferior to other methods of factorisation and are quite impracticable for larger numbers. It must also be noted that in testing a given number N for possible prime factors, it is not necessary to try a divisor greater than root N since if there is a factor larger than this there must also be a smaller one which has already been revealed.

Since testing by direct division by the primes in sequence is apparently only really applicable to numbers already covered in published tables, it would seem there is little point in pursuing this method further. However in one of the most successful methods of factorisation it is of the utmost importance to determine the highest possible limit of primes which are not factors of the number under examination. Apart from this it will be obvious that if we have eliminated all the primes less than the cube root of a given number N then there can be at most two factors of N and one of these must be between the cube root of N and the square root of N.

The best way of testing a number of small prime divisors is to employ Euclid's Algorithm for finding the HCF of two numbers. In essence this depends on the fact that if two numbers have a factor in common then the remainder after dividing one by the other will also contain the factor. Continuing in this manner the position is finally arrived at where either the remainder rn equals 1, in which case both numbers are relatively prime, or rn-1 is a multiple of rn. In the latter event the integer rn is the greatest common factor of the two numbers.

As an example taking the numbers 21 and 56 we divide thus: 56/21 = 2, + 14; 21/14 = 1, + 7; 14/7 = 2 exactly and therefore the last divisor -7- is the HCF of the two numbers 21 and 56.

We will discuss this algorithm in greater detail in Part 2.

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