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Dr. Katz DEq Homework Solutions 93

Dr. Katz DEq Homework Solutions 93 - either( l>(b...

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Algebraic Solutions of Differential Equations 93 If a~Z, a<0, then --1 Rx,(_a),,. 1 for all p sufficiently large. Then for P any A invertible mod N, the limit formula gives ( 1 > (bA) > (cA) >0. Just as above, replacing A by -A shows (bA)= (cA) for all invertible A modulo N. Hence c-b lies in Z, hence (as a~Z), c-b-a lies in Z, another contradiction, so ( does not occur. Thus we have shown that ar bq~Z, and that for every A invertible mod N, we have either ( l >(b A) > (cA) > (aA) >O or ( 1 > (aA) > (cA) >= (b A) > O. We next show that c-aq~Z and c-bq~Z. As by hypothesis a-br we cannot have both c- a ~ Z and c- b ~ Z. Suppose that c- a ~ Z. Then c-br hence for all A invertible mod N, A c-A br whence we have
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Unformatted text preview: either ( l > (b A) > (cA)=(aA) >O Or" ( 1 > (aA) = (cA) > (bA) >0. But ( cannot hold for both A and -A. Hence there exists a A invertible modulo N for which ( holds. Rewriting ( for this A via the limit formula (, we get ( 1> lim 1Rp(-a)= lim LRp(--c)> lim 1---Rp(-b)>0. p~ p p~ p p~ p pd -~ (N) pA =- 1 (N) pA =. 1 (N) Thus for all p sufficiently large with pA = 1 (N), we have ( 1 Rp(_C)>L Rp(_b)>O" P P This is incompatible with the first of the two following inequalities, one or the other of which holds for any sufficiently large prime in virtue of (6.4.0) ( l~ Rv(-b)>lRp(-c)>lRp(-a), P P P ( 1Rp(-a)> l~ Rp(-C)>__lRp(-b). P P P...
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