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

ELEMENTARY PROPERTIES OF INTEGERS

23

and show that the two representations are identical. We
have
aonh+.. +ah-ln+ah=bonk+.. +b75c-in+bk.
Then
aon+. * +ah-ln —(bon+... +bc —ln)=bk5-ah.
The first member is divisible by n. Hence the second is also.
But the second member is less than n in absolute value; and
hence, in order to be divisible by n, it must be zero. That is,
b =ah. Dividing the equation through by n and transposing
we have
aonh-l+ +ah-2n-(bon-+l+... +bk-2n) =bk -— ah-1
It may now be seen that bk-1=ah-1. It is evident that this
process may be continued until either the a's are all eliminated
from the equation or the b's are all eliminated. But it is
obvious that when one of these sets is eliminated the other is
also. Hence, h=k. Also, every a equals the b which multiplies the same power of n as the corresponding a. That is,
the two representations of m are identical. Hence the representation in the theorem is unique.
From this theorem it follows as a special case that any positive integer can be represented in one and in only one way in
the scale of io; that is, in the familiar Hindoo notation. It
can also be represented in one and in only one way in any other
scale. Thus
I20759=.76 +.75 + I.74+2.73+o.72+3.71+2.
Or, using a subscript to denote the scale of notation, this may
be written
(I20759)10= (1012032)7.
For the case in which n (of theorem I) is equal to 2, the
only possible values for the a's are o and I. Hence we have
at once the following theorem:
II. Any positive integer can be represented in one and in only
one way as a sum of different powers of 2.