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42 SPECIAL TOPICS IN THEORETICAL ARITHMETIC

100. Th. If a, ro, 11, 72, 73, =+ are as in § 99, then one and
only one set of integers ao, a1, @z, ***, G, ***, Qn, all positiv or
zero, can be found, such that

Qg <1’1, ay <7’2, asg <7’3, Scay Tk <7’k+1, siacs 5 <1’n+1

and a i {aleI (rm)}

=0 m=0

= Qoro + a17o?1 + Qo172 + o0 T Qutolifar3 v 0 Py

That one such set of integers can be found follows from § 99.

To show that only one such set can be found, suppose that
Qo, @1, Qs, **+, a3 is any such set.

That is, suppose that a,, a1, s, **+, as is any set of 7 4 1
positiv or zero integers, where # = 0, such that

Qg < 1, Q) < T2 Q2 < T3y **°, O < Thy1, **°y O < Tat1

and a = fh'_, {all'll (rm)}

=0 m=0
= ag’o + airor1 + Qarori?as + o0+ awrorirers T
From this set of & + 1 a’s and the given 7’s compute a set
of & 4+ 2 integers
gn+1y, Qny GQr—1y Qn—2y *°**y Qk—1, *°°y 43, G2, q1, Qo
as follows:
Let ¢g241 =0
gn = Qp + qr+1?n41

Gn=y"=Qp—1 + gn’h
gh—2 = Qp—2 + qr—1"h—1

qx = ar + Quia?it1, where 1L =k =h + 1
Gr—1 - ar—1 + Q%

gs = Qs -+ qara
q> = Qo + qs73
Q1 = a1 + Qare
go = @0+ i1