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et i AR T B

30 SPECIAL TOPICS IN THEORETICAL ARITHMETIC

By the hypothesis e = qo= @+ qin1
di — a3 4 Qate
qz = as + qsrs3

Gi—1 = Q1 & qr’k
g = Qr + Qr+17rp1

We wish to prove that

k—1 l k
g=3 {azH (rm)} + ¢ IT ()
=0 0 m=0

m=

for all values of k from 1 upward. that is,

- putting k£ = 1,

0 1 1
a=7> {Gz II (fm)} + ¢ II (rm) = aoro + quror1 = a0 + i1
=0 0 m=0

m=

putting k=2,

-t

l 2
a = { a; 11 (rm)} + @ I (rm) = aoro + arrors + garorira
0 m=0 m=0

~
I

putting 2 = 3,

a = aoro + Gi7or1 + Qeronira + Qstorirers

Of these statements the first is evidently tru by hypothesis.
Assume that the kth statement is tru, that is, that

l k
a = {az 11 (7'm)} -+ gk II (rm)
m=0 m=10
Then, since gx = ax + Gr+17r41,

k—1 l k
a = {az 1I (m)} + (ar + @er7e+r) 11 (7m)
0

m=

k—1 1 k k
%= {al II (m)} + ai II (m) + @rea?i4a I (7)
m=0 0

m=0 m=