B~ o N SR iy e =

P e e
S e i

3

W
i
1
1
5
i i

AE

‘:1‘.
il
:{':
|
i
|

& o

 

T e e = e

196 SPECIAL TOPICS IN THEORETICAL ARITHMETIC

Then d =¢t+ X (r — 1)r*! 4 s7», where we have the
k=2
middle terms if » > 1, but not if » = 1.
Let d’ be the result of interchanging the first and last digits
of d.

Thend =s+ Y (r — 1)kt 4 gy»
k=2
Hence

d—I—d’=(t—l—s)—{—Zi(r—l)rk—1+(s+t)r"
k=2
=(1’—1)—|—2(r—1)£rk—1—{—(r—1)rn
k=2
=<r—1>(1+2irk—1+1><rn)
k=2
Now suppose m = 2.
Thenn=1landd+d' =Fr—-1)14+N=r—1)r+@F—-1)

Next suppose m > 2.
Then » > 1 and

d+d = (=1 +2r 1) X P+ (= Dre
==+ =240 X4 (= D
= =D+ =D T trE 4 o= D
= (r— 1)+(r—2)§r’°+é2r’°+(r— 1r

=(r—1) +(r—2>[r+grk]+§2rk+rn+(r—1>rn,