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(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)?

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Please give a mathematical solution step-by-step, including all the patterns, conjectures, proofs and please take into consideration all the parts a, b, c, d in the problem

Let Z denote the set of integers. If m is a positive integer, we write Z, for the system of "integers modulo m." Some authors write Z/mZ for that system. For completeness, we include some definitions here. The system Zm can be represented as the set {0, 1,..., m – 1} with operations e (addition) and o (multiplication) defined as follows. If a, b are elements of (0, 1,..., m – 1}, define: a eb = the element e of (0, 1,..., m – 1} such that a +b - e is an integer multiple of m. a ob = the element d of (0, 1,..., m – 1} such that ab – d is an integer multiple of m. For example, 3e 4 = 2 in Z5, and 303 = 1 in Z4. To simplify notations (at the expense of possible confusion), we abandon that new notation and write a +b and ab for the operations in Zm, rather than writing ae b and a o b. The sequence of 2-powers is (2") = (2', 2°, 2°, 2ª, .). Evaluating that sequence in Z10, Z2s, and Z48, we find (2") = (2, 4, 8, 6, 2, 4, 8, 6, ...) (in Z10). (in Z2s). (in Z4s). (2") = (2,4, 8, 16, 4, 8, 16, ...) (2") = (2, 4, 8, 16, 32, 16, 32, ...) Compute (2") in Z„ for several other numbers m. What patterns do you observe? Here are some questions to guide your work. (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)? (d) Make some conjectures about the patterns mentioned in (b) and (c) above. Investigate other patterns that such sequences in Zm seem to satisfy, and make some conjectures. Can you prove some parts of your conjectures?
Transcribed Image Text:Let Z denote the set of integers. If m is a positive integer, we write Z, for the system of "integers modulo m." Some authors write Z/mZ for that system. For completeness, we include some definitions here. The system Zm can be represented as the set {0, 1,..., m – 1} with operations e (addition) and o (multiplication) defined as follows. If a, b are elements of (0, 1,..., m – 1}, define: a eb = the element e of (0, 1,..., m – 1} such that a +b - e is an integer multiple of m. a ob = the element d of (0, 1,..., m – 1} such that ab – d is an integer multiple of m. For example, 3e 4 = 2 in Z5, and 303 = 1 in Z4. To simplify notations (at the expense of possible confusion), we abandon that new notation and write a +b and ab for the operations in Zm, rather than writing ae b and a o b. The sequence of 2-powers is (2") = (2', 2°, 2°, 2ª, .). Evaluating that sequence in Z10, Z2s, and Z48, we find (2") = (2, 4, 8, 6, 2, 4, 8, 6, ...) (in Z10). (in Z2s). (in Z4s). (2") = (2,4, 8, 16, 4, 8, 16, ...) (2") = (2, 4, 8, 16, 32, 16, 32, ...) Compute (2") in Z„ for several other numbers m. What patterns do you observe? Here are some questions to guide your work. (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)? (d) Make some conjectures about the patterns mentioned in (b) and (c) above. Investigate other patterns that such sequences in Zm seem to satisfy, and make some conjectures. Can you prove some parts of your conjectures?
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