Unformatted text preview: theory, deduce the value of p . 6 (7 points) . The continued fraction expansion of √ 5 is h 2 , 4 , 4 , . . . i . If h 2 , 4 , 4 , . . . , 4  {z } 99 4 s i = h/k (in lowest terms), calculate h 25 k 2 . 7 (5 points) . Prove that there are an inﬁnite number of primes congruent to 3 mod 4. 8 (6 points) . Suppose that p = a 2 + b 2 , where p is an odd prime number and a is odd. Show that ± a p ² = +1. (Use the Jacobi symbol.) 9 (8 points) . Let a and b be positive integers. Show that φ ( ab ) φ (gcd( a, b )) = φ ( a ) φ ( b ) gcd( a, b ) , φ = Euler φfunction . (Example: If a = 12 and b = 8, the equation reads 32 · 2 = 4 · 4 · 4.) 10 (5 points) . Find all solutions in integers y and z to the equation 6 2 + y 2 = z 2 ....
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This note was uploaded on 10/31/2009 for the course STAT 131A taught by Professor Isber during the Spring '08 term at Berkeley.
 Spring '08
 ISBER

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