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Unformatted text preview: 23 (thus this includes both the primitite and nonprimitive ones). 5. (5 pts) Find the set of all primes p 6 = 2 , 5, for which x 2 10 mod p is solvable. 6. (6 pts) Compute the quadratic irrationality represented by the periodic continued fraction h 2 , 5 i and h 3 , 4 i 7. (6 pts) Compute the continued fraction expansion of 11 and 30. 1 2 MATH 115 PRACTICE FINAL EXAM 8. Let p be a prime of the form p = a 2 + b 2 , with a , odd, b even. Note that in this case, p 1 mod 4. a) (1 pts) Show that gcd( a,b ) = 1. b) (2 pts) Show that: p a = 1 c) (1 pts) Hence use the quadratic reciprocity law for Jacobi symbols, together with the results of part b), to show a p = 1...
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This note was uploaded on 06/15/2009 for the course MATH 115 taught by Professor Mok during the Fall '07 term at University of California, Berkeley.
 Fall '07
 MOK
 Number Theory, Integers

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