148 SPECIAL TOPICS IN THEORETICAL ARITHMETIC

366. Th. If the modulus is r™ + 1,

e e R Xk
k=0 (mod 2) k=1 (mod 2)
or a =Y p,(keven) — 3 pr (kodd)
or a=potpetpst-)—(r+pst+pst---)

In words: If the modulus is ™ + 1, any number is congruent
to the difference of the sums of its two sets of alternate m-figure
periods, the minuend containing the period that contains the units
digit.

For example, 231,573,024 = (24 + 231) — 573 (mod r™ + 1)

m—1
367. Th. a = > [( o ar — -3 a;\.) rl]
=0 k=l (mod 2m) k=l4+m (mod 2m)

(mod r™ + 1)

For Po=a+ ar+ - + auar™? § 313
P2 =l ons + QA2m1? + g _|_ agm_lrm—l
o - Qi+ Ginia? + - + Bt

l

Adding columnwise,

pot 2t pit - = (@0 + on+ aim+ )
+ (al + a’2m—|—l + aw.lm—fi\—l + A )7’
+ i
+ (am—-l —I— A3m—1 + Csm—1 + .- )rm~1
= Z ak+ ( Z ak> 7_|_ TR +( Z a//r) pym—1
k=m—1 (mod 2m)

k=0 (mod 2m) k=1 (mod 2m)

m—1
=0 k=l (mod 2m)

m—1
Similarly p1 +ps+ s+ - -+ = 2 [( >3 (Zk) rl]
k=l+m (mod 2m)

=0