Unformatted text preview: 6 . Find the number of primitive roots mod p 2 when p is the prime 257. 7 . Express the continued fraction h 6 , 6 , 6 , . . . i in the form a + b √ d , with a and b rational numbers and d a positive non-square integer. 8 . 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 . Let n be an integer. Show that n is a diﬀerence of two squares (i.e., n = x 2-y 2 for some x, y ∈ Z ) if and only if n is either odd or divisible by 4. 10 . Let n be an integer greater than 1. Prove that 2 n is not congruent to 1 mod n ....
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- Spring '08
- Prime number, Rational number, Smallest prime factor, K. A. Ribet, Professor K. A. Ribet