The theory of numbers, by Robert D. Carmichael ...

OTHER TOPICS

83

development of the theory of these forms has been given by
the present author in a memoir published in 1913 in the Annals
of Mathematics, vol. 13, pp. 30-70.
~ 44. ANALYTICAL THEORY OF NUMBERS
Let us consider the function
P(x)=- —, I x p< I.
In (i-X2t)
k =0
It is clear that we have
P(x)= n -1 ni (I+X2+X22- +x3'2+...)
k=O I —X k =O
oo
= z G(s)x,
s=0
where G(o) = i and G(s) (for s greater than o) is the number of
ways in which the positive integer s may be separated into like
or distinct summands each of which is a power of 2.
We have readily
00 00
(I-X) G(S)X = (I -x)P(x) =P(X2) = 2 G(S)X2;
s=0 s=0
whence
G(2s + )= =G(2)=G(2s-I) +G(s), (A)
as one readily verifies by equating coefficients of like powers
of x. From this we have in particular
G(o)=i, G(I)= I, G(2)=2, G(3)-2,
G(4)=4, G(5)=4, G(6)=6, G(7)=6.
Thus in (A) we have recurrence relations by means of which
we may readily reckon out the values of the number theoretic
function G(s). Thus we may determine the number of ways in
which a given positive integer s may be represented as a sum
of powers of 2.