I

S R e e e e e e "

e —

s S e

q‘
|
|

A B

e SR AR oy AT o 73 T Bt AT

 

32 SPECIAL TOPICS IN THEORETICAL ARITHMETIC

If 2 > 1, our conclusion may be stated:

a = ao + air1 + aarire + asrirers + ¢ -
+ apanirars c o i1 + qiriters * * * Tk

The term a;_i717s7s *  + 71,1 however has no sense if & = 1.
By introducing the factor 7o, which equals 1, we avoid this
difficulty. The term aj_i7¢717o7s « + + 7,1 has sense whether %
equals 1 or is greater than 1.

Our conclusion can also be put in the form

a=(ao+ (a1 + (a2 + (@s + - -
+ (ak—z > (Gk—1 -+ Qkfk)f’k—l)f’k—z s ')7‘3)7’2)7’1)7’0

or the form

a = qi’x + a«k——l)rk—l = ak—2)7'k_2 oo

+ as)rs + az)r: + a1)r1 + ao)ro
which, in case £ > 1, may be written

a=a0+ (@1 + (a2 + (as+ -~
3 _(ak—z + (ar—1 + Qu?r)7k—1)7k—2 * * *)73)72)"1

or
a = Qi+ ak—l)rk—l + ak_z)f’k—z g anist 03)7'3 <+ (12)7’2 =+ 01)7’1 + ao

When we have performd the work indicated in the hypothe-
sis of this theorem, we have as a result the number a exprest
in the form

aoro + arrory + asrorire + - - -+ @rarorare c cc Tr—1 T+ Qolar2 Tk
With reference to this form the numbers
Qo, A1, A2, ***, Qk—1, Gk
may be called the digits of the notation.

Ex. 1. Suppose a = 5; let the series of divisors be four in
number, 71, 72, 73, 74, respectivly equal to 4, 0, — 2, 3; for the
quotients ¢i, g2, ¢s, ¢4 choose the values 2, — 2, 0, 2.