571p03 - p ) when p − 1 | k . 4. (Charles Dodgson) In a...

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MATH 571 ANALYTIC NUMBER THEORY I, SPRING 2011, PROBLEMS 3 Due 1st February 1. (i) Let m N . Prove that ( y 1)( y m 1 + y m 2 + ··· + y + 1) = y m 1 . (ii) Let n N . Prove that ( x 2 + 1)( x 2 1)( x 4 n 4 + x 4 n 8 + ··· + x 4 + 1) = x 4 n 1 . (iii) Let p be a prime number with p 1 (mod 4). Prove that x 2 ≡− 1(mod p ) has exactly two solutions. 2. (i) Find the exponents to which 2, 3 and 5 belong modulo 23. (ii) Find a primitive root g modulo 23, construct a table of indices (that is, for each y with 0 y 21 list the x with 1 x 22 for which g y x (mod 2)3), and solve the congrence x 6 4 (mod 23). 3. Show that 1 k +2 k + ··· +( p 1) k 0(mod p )when p 1 ° k and is ≡− 1 (mod
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Unformatted text preview: p ) when p − 1 | k . 4. (Charles Dodgson) In a very hotly fought battle, at least 70% of the combatants lost an eye, at least 75% an ear, at least 80% an arm, and at least 85% a leg. What can you say about the percentage that lost all four members? 5. (P. T. Bateman) Would you believe a market investigator who reports that of 1000 people, 816 like candy, 723 like ice cream, 645 like cake, while 562 like both candy and ice cream, 463 like both candy and cake, 470 like both ice cream and cake, while 310 like all three?...
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This note was uploaded on 01/23/2012 for the course MATH 571 taught by Professor Vaughan,r during the Spring '08 term at Pennsylvania State University, University Park.

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