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Unformatted text preview: a/p ) = (-1) n ). (a) (8 / 11) (b) (7 / 13) (c) (5 / 19) (7) (a) Let p be an odd prime, and suppose that a is an integer with ( a, p ) = 1. Show that the Diophantine equation x 2 + py + a = 0 has an integral solution if and only if (-a/p ) = 1. (b) Determine whether or not the Diophantine equation x 2 + 7 y-2 = 0 has a solution in the integers. (8) Prove that 2 is not a primitive root modulo any prime of the form p = 3 2 n +1, except when p = 13. (9) For a prime p 7 (mod 8), show that p | (2 ( p-1) / 2-1). 1 2 MATH 115B PROBLEM SET I APRIL 5, 2007 (10) (a) Suppose that p is an odd prime, and that a and b are integers such that ( ab, p ) = 1. Prove that at least one of a , b or ab is a quadratic residue modulo p . (b) Show that, for some choice of n > 0, p divides ( n 2-2)( n 2-3)( n 2-6) ....
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