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Algebraic Solutions of Differential Equations
91
Asp>l~[, p~
N
1
<
,andhence
(6.5.2.1.2) o____
\W/)N < 1
whence
(6.5.2.1.3)
(6.5.3)
pN
.
Q.E.D.
Corollary.
Let ~ ~ Z, N ~ Z with N ~ O. For each invertible element
A in Z/N
Z,
we have the limit formula
(6.5.3.1)
limit P
p
pd=l(N)
0
if
~~Z
~~Z, ~<0
if ~~Z, ~>0.
Proof
If
a/NCZ,
this follows immediately from (6.5.2.1). If
~/NeZ,
then for all primes p>
I~1,
pRp
=
~
=
~
if N>0.
Np
6.6.0.
Proposition.
Let a, b, c~Q, and suppose that none of the "exponent
differences" 1 c, cab, ab lies in Z. In order that
(6.2.6)
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This note was uploaded on 12/21/2011 for the course MAP 4341 taught by Professor Normankatz during the Fall '11 term at UNF.
 Fall '11
 NormanKatz

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