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prob1

# prob1 - a/p =-1 n(a(8 11(b(7 13(c(5 19(7(a Let p be an odd...

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MATH 115B PROBLEM SET I APRIL 5, 2007 (1) Prove that the quadratic congruence 6 x 2 + 5 x + 1 0 mod p has a solution modulo every prime p , even though the equation 6 x 2 + 5 x + 1 = 0 has no solution in the integers. (2) Show that 3 is a quadratic residue modulo 23, but is a non-residue modulo 31. (3) Given that a is a quadratic residue modulo the odd prime p , prove the following: (a) a is not a primitive root of p . (b) The integer p - a is a quadratic residue or non-residue modulo p according as p 1 (mod 4) or p 3 (mod 4). (c) If p 3 (mod 4), then x ≡ ± a ( p +1) / 4 (mod p ) are the solutions of the congruence x 2 a (mod p ). (4) If p = 2 k +1 is a prime, show that every quadratic non-residue modulo p is a primitive root modulo p . (5) Find the value of the following Legendre symbols: (a) (19 / 23) (b) ( - 23 / 59) (c) ( - 72 / 131) (6) Use Gauss’s Lemma to compute each of the following Legendre symbols (i.e., in terms of the notation that we used in class, ﬁnd the integer n in Gauss’s Lemma for which (

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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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prob1 - a/p =-1 n(a(8 11(b(7 13(c(5 19(7(a Let p be an odd...

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