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NT2009HW6

NT2009HW6 - 4400/6400 PROBLEM SET 5 PETE L CLARK A sucient...

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4400/6400 PROBLEM SET 5 PETE L. CLARK A sufficient number of problems: 4 for 4400 students, 5 for 6400 students. 5.1) Evaluate these Legendre symbols (the denominators are all prime numbers): 85 101 , 29 241 , 101 1987 , 31706 43789 . 5.2) Make up another six Legendre symbol problems, with numbers ranging from 3 to 5 digits (for this you will need to do something to find some primes in this range, but for instance you can find lists of primes on the internet), and solve them. Try to find the approximate range in which it becomes difficult to do the computations directly in terms of Legendre symbols – i.e., by factoring at every stage – and hence it becomes preferable to use the Jacobi symbol. 5.3) For each of the following integers N , find all primes p such that N is a square modulo p : a) 31. b) 2007. c) The year of your birth. 5.4) Prove that the quadratic reciprocity law is equivalent to: for distinct odd primes p, q , p q = ( - 1) q - 1 2 q p . 5.5) Prove Lemma 6 of [Quadratic Reciprocity II: The Proofs].

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