194 SPECIAL TOPICS IN THEORETICAL ARITHMETIC

Thus
a + i P o (ao RS ‘2 ap_r* ! + a,,r")
k=1 k=2
- (r S r-r"‘l)
k=2
= (r + ay) + X (r + ar_)r* 1 + a.r”
k=2
And
a + i rh = (an + i Gpar*! 4 do””)
k=1 k=2
‘\::'1 + i rk—l + r‘n)
1 k=2
| = a, + i (ar—1 + )71 4 (ag + Dr»
k=2
Hence

a—a =[r— (a,—an)]+ é2(r— D7t 4 [(an — ag) — 1 "

Set s = (an —ay) — 1
and t=r — (a, — ao)
Thens+t=7r —1
and

I

a—a =t+ Y (r — Drkt 4 s
k=2

Since a, and a, are digits and a, > a,

0=a<a.=r—-1

Since a, > @y, a, — ay > 0

Hence an — e =1
Since ¢, =7 — 1
and a = 0,

an——a0<r-—-1