Unformatted text preview: P + p d Q with P; Q 2 Z and Q j ( d & P 2 ) . 5. Let d 2 Z be positive and not a square. Let P = 0 ; Q = 1 ; ² = p d; q = [ ² ] and consider the recursion P n +1 = Q n q n & P n ; Q n +1 = d & P 2 n +1 Q n ; ² n +1 = P n +1 + p d Q n +1 ; q n +1 = [ ² n +1 ] : Prove that there are an in±nite number of n such that Q 2 n = 1 . (Hint: You should review the material on Pell&s equation for this problem.) 1...
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 Winter '09
 N.R.Wallach
 Math, Number Theory, Rational number, Continued fraction, Primitive root modulo n, Dirichlet character

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