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math104bPracticemid

# math104bPracticemid - P p d Q with P Q 2 Z and Q j d& P...

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Practice Problems for the Midterm 1, Let m 2 Z and m > 1 : Show that if a 2 Z and gcd( a; m ) = gcd( a ° 1 ; m ) = 1 then 1 + a + a 2 + ::: + a ° ( m ) ° 1 ± 0 mod m: (Here ° ( m ) is Euler°s ° -function, as usual.) 2. Let p be a prime. Recall that the principal Dirichlet character modulo p is given by ± 0 ( n ) = ° 1 if p - n 0 if p - n : For what primes p does there exist a non-principal Dirichlet character ± such that ± 3 = ± 0 ? 3. Find the smallest prime p such that there is an integral solution to x 2 ° x ° 1 ± 0 mod p . 4. Let ² be the number with simple continued fraction [ m; m; m; m; ::: ] (i.e. q j = m all j ). Write ² in the form
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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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