hw_03 (1)

hw_03 (1) - ence f ( x ) 0( p ) has n solutions a 1 , a 2 ,...

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Number Theory, Homework 3rd batch due: Wednesday, February 11 All exercises are worth 5 points. Please select five problems since there is a cap of 25 points. If you decide to do more than five problems, please indicate which ones you want graded since otherwise the five lowest scores will be selected. Sometimes, only five problems will be assigned, in which case the selection issue is moot. Exercise 1. 2.7.2: Prove that 2 x 3 + 5 x 2 + 6 x + 1 0(7) has three solutions mod 7. Exercise 2. 2.7.3: Show that every integer satisfies x 14 + 12 x 2 0(13). Exercise 3. 2.7.4: Let f ( x ) be an integer polynomial and assume that the congru-
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Unformatted text preview: ence f ( x ) 0( p ) has n solutions a 1 , a 2 , . . . , a n mod p . Show that there is an integer polynomial q ( x ) such that the congruence f ( x ) ( x-a 1 )( x-a 2 ) ( x-a n ) q ( x )( p ) holds for each integer. Exercise 4. 2.7.6: Show that Theorem 2.26 becomes false when mod p is replaced by mod m where m is a composite number. Exercise 5. 2.7.10: Let p be an odd prime and write the rational number 1 1 + 1 2 + 1 3 + + 1 p-1 as a fraction a b reduced to lowest terms, i.e., ( a, b ) = 1. Show that p divides evenly into a ....
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This note was uploaded on 04/05/2010 for the course CS 150 taught by Professor Humphreys,g during the Spring '08 term at UVA.

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