Unformatted text preview: . Let p be a prime that is congruent to 1 mod 4. View the quadratic residues (i.e., squares) mod p as integers between 0 and p1. Show that the sum of these integers is p ( p1 4 ). [Example: when p is 5, the residues are 1 and 4. Their sum is 5.] 8 (4 points) . If eggs in a basket are taken out 2, 3, 4, 5 and 6 at a time, there are 1, 2, 3, 4 and 5 eggs left over, respectively. If they are taken out 7 at a time, there are no eggs left over. What is the least number of eggs that can be in the basket? 9 (5 points) . Show that a positive integer n is a perfect number if and only if X d  n 1 d = 2. If n is a perfect number, show that tn is not a perfect number when t > 1. 10 (6 points) . Let n be a positive integer. Suppose that the Fermat number p = 2 2 n + 1 is prime. Prove that 3 ( p1) / 2 ≡ 1 mod p ....
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 Spring '08
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 Prime number, positive integer

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