(a) That sequence (2") appears to repeat after a few initial terms. Why must such repetition occur for every m? (b) Let p(m) be length of the "tail" of terms that occur before the repeating part begins. For instance, p(10) = 0, p(28) = 1 and p(48) = 3. For which m does p(m) = 0? When does p(m) = 1? When does p(m) = 2? (c) Let o(m) be the period (length of the periodic cycle) of the sequence (2") in Zm- For instance, o(10) = 4 and o(28) = 3 and o(48) = 2. How is o(m) related to o(2m)? What about o(m) and o(3m)? How is o(35) related to o(5) and o(7)?
(a) That sequence (2") appears to repeat after a few initial terms. Why must such repetition occur for every m? (b) Let p(m) be length of the "tail" of terms that occur before the repeating part begins. For instance, p(10) = 0, p(28) = 1 and p(48) = 3. For which m does p(m) = 0? When does p(m) = 1? When does p(m) = 2? (c) Let o(m) be the period (length of the periodic cycle) of the sequence (2") in Zm- For instance, o(10) = 4 and o(28) = 3 and o(48) = 2. How is o(m) related to o(2m)? What about o(m) and o(3m)? How is o(35) related to o(5) and o(7)?
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter1: Fundamentals
Section1.1: Sets
Problem 10E
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