prob2 - k + 1. [Hint: Assume that p 1 , p 2 , . . . , p r...

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MATH 115B PROBLEM SET II APRIL 17, 2007 (1) Find the value of the following Legendre symbols: (a) (71 / 73) (b) ( - 219 / 383) (c) (3658 / 12703) [Hint: 3658 = 2 · 31 · 59.] (2) Prove that 3 is a quadratic non-residue of all primes of the form 2 2 n +1, and all primes of the form 2 p - 1, where p is an odd prime. [Hint: For all n , 4 n 4 (mod 12).] (3) Verify that if p is an odd prime, then ± - 2 p ² = ³ 1 if p 1 (mod 8) or p 3 (mod 8) - 1 if p 5 (mod 8) or p 7 (mod 8) (4)(a) Prove that if p 3 is an odd prime, then ± - 3 p ² ³ 1 if p 1 (mod 6) - if p 5 (mod 6) (b) Using part (a), show that there are infinitely many primes of the form 6
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Unformatted text preview: k + 1. [Hint: Assume that p 1 , p 2 , . . . , p r are all the primes of the form 6 k + 1, and consider the integer N = (2 p 1 p 2 p r ) 2 + 3.] (5) Determine whether the following quadratic congruences are soluble: (a) x 2 219 (mod 419). (b) 3 x 2 + 6 x + 5 0 (mod 89). (6) Determine which primes can divide integers of the forms n 2 + 1, n 2 + 2 or n 2 + 3 for some value of n . (7) Find all solutions of the quadratic congruence x 2 11 (mod 35) . 1...
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