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hmwk_6 - p 1 and p 2 be prime Is p 1 × p 2 1 always prime...

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Discrete Structures Sept 26 2011 Assignment 6: Due at beginning of class Monday, Oct 10 Prof. Hopcroft *******Note: this homework is subject to the style guide on the website. Points will be deducted for homeworks not following the guidelines.******** *******Please print out and staple the grade sheet to the back of your homework!****** 1. If, for a given a there exists x such that ax 1 mod m does that imply that a and m are relatively prime? Prove why or why not. 2. Let p > 2 be a prime. Then an integer i [1 , p - 1] is a perfect square if there exists, j such that j 2 i mod p . Prove the number of perfect squares is p - 1 2 . Consider the following hints: Show that x 2 ( p - x ) 2 mod p . Show that for x, y [1 , p - 1 2 ], if x 6 = y , then x 2 is not equivalent to y 2 mod p 3.
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Unformatted text preview: p 1 and p 2 be prime. Is p 1 × p 2 + 1 always prime? Give a proof or counter example. (b) For some prime p , is p-1 (mod p ) ever a perfect square? 4. Show that there is always a one-to-one and onto mapping between any two countably infinite sets. Make sure to use a well defined function as your mapping, ie the identity mapping, for some set S , f : S → S where f ( s ) = s . 5. We want to show that the set of reals and the set of pairs of reals have the same cardinality. Note we cannot list the reals so we can’t talk about the i th real mapping to the j th real pair. Instead, construct a one-to-one and onto mapping, f : R → R 2 . 1...
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