‘PRIMES AND FACTORS’ 47

after 11, and so on, were crossed out. When the process was
complete, the numbers remaining uncrossed were found to
be primes. Since this is only a selective method and the num-
ber of primes is infinite, the Sieve has only a limited use and
gives no clues to a general formula.

The primes less than 100, shown in their respective groups
of ten consecutive numbers are:

Between 1 and 10 I 2 3 5 il
. 11 and 20 11 13 17 19
5 21 and 30 23 29
% 31 and 40 31 37
i 41 and 50 41 43 47
4 51 and 60 53 59
,, 61and %0 61 67
L 71 and 8o 71 73 79
,,  81andgo 83 89
,, 91 and 100 97

It is at once apparent that these numbers do not form any
series similar to those we have already met, and are seem-
ingly not related to each other in any fixed way at all.

An ingenious method of finding the factors (if any) of a
number was devised by Fermat, who based his calcula-
tions on the fact that the factors of a?—b? are (a-b) and
(a—b). Thus, if any number can be expressed as the differ-
ence between two squares, then its factors may be readily
discovered.

The procedure is to calculate the approximate square root
of the number to be factorized. The next largest integer is
then taken as the starting point. If this integer be called g,
and the original number be called JV, then if a*— N is another
perfect square (say b2), it follows also that a*—5*=N. By this
means, the original number VN is expressed as the difference
between two squares.

It will be seen that a2 is a certain square number slightly
greater than N, but if a>—JV is not a perfect square then we
can substitute (¢+1)2 or (a+2)? or, in fact, any square
larger than a® as we proceed. Thus, if 22—V is not a perfect
square, we proceed to substitute (¢+1) and progressively

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