‘PRIMES AND FACTORS’ 57

Not derived
2,3, 5, 6, 7, 10, 11, 13, 14, 17, 18, 19, 22, 23, 26, 29, 30,
31, 34, 37, 38, 41, 42, 43, 46, 47, 50

It will be seen at once that not all composite numbers can
be derived from the odd series, but those which are excluded
can be built up from a different series—that of the con-
secutive even numbers, 2, 4, 6, 8, etc.

We therefore have a method of sorting the primes from
the composites; it is in fact a new kind of sieve. If a number
cannot be represented either as the sum of a number of
consecutive odd or even numbers, then it must be a prime.

This may be more easily appreciated from the following

table which shows the sieve in operation:

Numbers which can be built up from: Numbers which
(a) Odd series (&) Even series cannot be
built up
1 6 2
4 Io 3
8 14 5
9 18 7
12 22 11
15 26 13
16 30 17
20 34 19
21 38 23
24 42 29
25 46 31
27 50 37
28 41
32 43
33 47
D0
36
39
40
44
45
48
49

All the numbers in the first two columns are composites
whereas all the numbers in the third column are primes. This
knowledge is, by itself, of little practical use, but it is never-
theless of great importance and leads to a new method

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