‘PRIMES AND FACTORS’ 49

as we employ it as a substitute for 4, all we need do is to add
progressively higher consecutive odd numbers to the value
of a®— N, until at some stage the total is a perfect square.

The method just outlined is useful where the difference
between the factors of a number is relatively small. Where
the difference is relatively large, the process is far too pro-
tracted.

Another method of factorizing numbers was devised by
Euler. This was based on the expression of a number as the
sum of two squares in two different ways. For example, the
number 221 can be expressed both as (102411%) and as
(52+4142). Not all numbers can be so expressed and, in any
case, the method is complicated and brings us no nearer to a
general solution.

Although it is not possible to identify every prime immedi-

ately, many facts about primes are known. It has, for in-
stance, been proved that the number of primes is infinite. The
proof, due to Euclid, is simple and depends upon the basic
fact that no two consecutive numbers can have any similar
factors. This is so because if x is exactly divisible by », then
x+1, when divided by », will give a remainder of 1.

If instead of x and x -+ 1 we substitute x! and (x!4-1) where
x!=1X2X3X4X5X ... Xx, the principle remains the
same. Therefore (x!+1) cannot have any factors common to
x! so that if (x!+41) has any prime divisors at all, then they
must be distinct from any number lower than x. Alternatively
(x!4-1) may itself be prime. One of these propositions must
apply so that, in either case, we have proof that there are
primes greater than x whatever value we take for x.

In addition to ordinary primes, numbers may be des-
cribed as relatively prime to each other where they have no
common factors. In the same way, the sum of any two rela-
tively prime numbers is also relatively prime to their differ-
ence as well as to the numbers themselves. For example, the
numbers 16 and 25 are relatively prime. Their sum is 41 and
their difference is 9, and the four numbers, 16, 25, 41 and g
are all relatively prime.

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