{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

math104bPracticemid

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

Info icon This preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
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
Image of page 1
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern