(a) Let p be an odd prime and gcd(a, p) = gcd(k , p) = 1. Show that if the equationx2 -
Chapter 9, Problem 16(choose chapter or problem)
(a) Let p be an odd prime and gcd(a, p) = gcd(k , p) = 1. Show that if the equationx2 - ay2 = kp admits a solution, then (a/ p) = 1; for example, (2 /7) = 1, because62 - 2 . 22 = 4 . 7.[Hint: If xo, Yo satisfy the given equation, then (x0yg2)2 =a (mod p).](b) By considering the equation x 2 + 5 y2 = 7, demonstrate that the converse of the resultin part (a) need not hold.(c) Show that,for anyprime p = 3 (mod8),the equation x2 - 2 y2 = p has no solution.
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