‘PRIMES AND FACTORS’ 55

However, Miller and Wheeler have now found larger ones,
the largest being 180(2%2" —1)24-1.

It can be shown that if 2 —1 is prime, then P must also
be prime. If we suppose that P is not prime then it must have
at least two factors, @ and 4, each of which is greater than 1.
Then 2 —1 becomes 2% —1, and this expression has the
factor 2% —1, which is neither equivalent to 1 nor to 2f—1,
whence P must be prime. It does not necessarily follow that
if P is prime then 2f—1 is also prime.

Numbers of the form 2” —2 are divisible by P if the latter
is prime. For a long time it was considered probable that,
if P were composite, then 2¥ —2 could not be divisible by P.
This was, however, eventually proved by Sarrus to be incor-
rect for the value P=341.

The subject of primality is not unconnected with the series
of consecutive odd numbers. To conclude this chapter, the
writer appends a brief survey of the results of individual
researches conducted in this connexion, establishing a rela-
tionship which is of interest if not of immediate practical use.

SERIES AND PRIMES

In the earlier chapters we have been largely concerned
with the use of the series of consecutive odd numbers in the
formation of ‘shape’ or ‘power’ numbers. This series is also
of importance in the construction of every number and has a
particular reference to the question of primality.

It was noted that, in order to build up the number 22, we
always required z consecutive numbers and that the same
requirement applied for n3. It may now be stated that any
power number of the form »#* can be built up from z con-
secutive odd numbers irrespective of the value of x. Where
n is odd, the central number in the odd series for »® is equi-
valent to n®~1. For 5% the central number will be 54-1 = 53
= 125, so that:

5i=121+412834125-+127 4129 =625
Where 7 is even, there is no single central number in the

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