CONGRUENCE 149

Then

a= (po+p2tpat--:)— (1 +ps+ps+ ---)(modrm§+1)
366.

m—1 m—1
-E (L zaep Bl
=0 k=l (mod 2m) =0 k=l4+m (mod 2m)
m—1
= [( 25 ity e 2. ak) r’]
=0 k=l (mod 2m) k=l+m (mod 2m)

This theorem might also be stated, writing the terms in
reverse order :

m—1
a=) [( 2 ar — o ak) r’"‘l_l]
=0 k=m—I1—1 (mod 2m) k=2m—1—1 (mod 2m)
(mod ™ + 1)

For example, if » = ¥ and m = 3,

115,342,789,437 = (7 — 8)100 4 (7 — 9)10 + (9 — 14) = — 125
(mod 10 4 1)

368. Th. By the process of § 364 or that of § 365 any number
whose digits are all positiv or zero, or all negativ or zero, and
which has more than one m-figure period may be reduced to a
one-period number with respect to the modulus r™ + 1.

Asin § 343 we have |a| > |po+p1+ P2+ -+ + i

But

[potp1+ ot Fpi=|po—pr+pa—ps+ -+ (—1)p]
Hence la] > [po—pr+p2— s+ -+ + (— 1)y

Thus applying the rule once reduces the given number
numerically.

If the new number po — p1 + po — p3+ -+ + (— 1) is
not a one-period number, then, if we change it, if necessary,
to a form in which its digits are either all positiv or zero, or
all negativ or zero, it may similarly be reduced in magnitude.

This process must, as in § 343, if continued, finally produce
a one-period number.