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Dr. Katz DEq Homework Solutions 91

Dr. Katz DEq Homework Solutions 91 - Algebraic Solutions of...

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Algebraic Solutions of Differential Equations 91 Asp>l~[, p~- N 1-- < ,andhence ( o____ \W-/--)-N -< 1 whence ( (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 ( limit P p pd=-l(N) 0 if ~-~Z if -~-~Z, ~<0 if ~-~Z, ~->0. Proof If a/NCZ, this follows immediately from ( If ~/NeZ, then for all primes p> I~1, -p-Rp = ~ = ~ if N>0. Np 6.6.0. Proposition. Let a, b, c~Q, and suppose that none of the "exponent differences" 1 -c, c-a-b, a-b lies in Z. In order that (6.2.6)
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