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Math 115 - Spring 1994 - Ribet - Final

Math 115 - Spring 1994 - Ribet - Final - Mathematics 115...

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Mathematics 115 Final Exam, May 18, 1994 Professor K. A. Ribet 1. Let A be a finite abelian group. If ϕ : A C * is a non-trivial homomorphism, show that a A ϕ ( a ) = 0. 2. Calculate the number of solutions to the congruence x 3 8 mod 5040. (Note that 5040 = 7!.) 3. A pseudoprime is a composite integer p for which 2 p 2 mod p . Show that 11 · 31 is a pseudoprime. Prove that every Fermat number 2 2 n +1 is either a prime or a pseudoprime. [The first known even pseudoprime 161038 was found by Berkeley’s D. H. Lehmer in 1950.] 4. Suppose that p is an odd prime and that a is an integer prime to p . Gauss’s lemma may be summarized as the identity a p = ( - 1) μ . Explain the definition of the quantity μ that appears in this formula. Use Gauss’s lemma to compute 2 p when p 1 mod 8. 5. Recall that ζ ( s ) = n =1 1 n s for s > 1. Prove that ζ ( s ) - 1 < 1 t - s dt < ζ ( s ) by using techniques from first-year calculus. Use these inequalities to prove that the limit
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