Mathematics 115Final Exam, May 18, 1994Professor K. A. Ribet1.LetAbe a finite abelian group. Ifϕ:A→C*is a non-trivial homomorphism, showthata∈Aϕ(a) = 0.2.Calculate the number of solutions to the congruencex3≡8 mod 5040.(Note that5040 = 7!.)3.A pseudoprime is a composite integerpfor which 2p≡2 modp. Show that 11·31 is apseudoprime. Prove that every Fermat number 22n+1 is either a prime or a pseudoprime.[The first known even pseudoprime161038was found by Berkeley’s D. H. Lehmerin 1950.]4.Suppose thatpis an odd prime and thatais an integer prime top. Gauss’s lemmamay be summarized as the identityap= (-1)μ. Explain the definition of the quantityμthat appears in this formula. Use Gauss’s lemma to compute2pwhenp≡1 mod 8.5.Recall thatζ(s) =∞n=11nsfors >1.Prove thatζ(s)-1<∞1t-sdt < ζ(s) byusing techniques from first-year calculus.Use these inequalities to prove that the limit
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