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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 y2 = 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 ( p1) / 21). 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 22)( n 23)( n 26) ....
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 Fall '09
 Congruence, Integers, quadratic residue, quadratic residue modulo

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