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14. Demonstrate that 21 has no primitive root.

14. Demonstrate that 21 has no primitive root.

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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9. The Liouville Lambda (A) function is a number theoretic function – similar to T,0,¢ et al. It is denaic by if n = 1 X(12) : (-1)kı+k2+...+kr if n >1 where n = p p...pr e N. = Pi P2 Prove that if n is a perfect square, then X(n) = 1. 10. Let r be a primitive root of some n 2 3. Prove that r ¢(n) = -1 (mod n). 11. Find 7(n), o(n), X(n), µ(n), w(n), and ø(n) for the following integers.' • 2250 • 199 • 286936650 • 22! 12. Let p = 17 and d be a divisor of o(p). Determine (d) for each d. List each element having order d, for all divisors, d, of (p). %3D 13. Calculate all the primitive roots of 41 and 26. 14. Demonstrate that 21 has no primitive root. 15. Let r be a primitive root of n. If gcd(a, n) = 1, then the smallest positive integer k such that a = rk (mod n) is called the index of a relative to r, denoted by ind,a. The theory of indices can be used to solve congruences. Consider the properties of indices (p. 164) and example 8.4 (p. 164). Solve %3D 8x = 11 (mod 13) using the fact that 6 is a primitive root of 13. Construct a table of indices for 6 modulo 13. 16. (7 points (bonus)) Solve the exercise that was previously skipped (since only 14 are required). Be sure to clearly mark the exercise you wish to count for extra credit. Here is an excellent online large number calculator that has a built-in modulo n feature: https://www.calculator.net/big-number-calculator.html 'µ(n) is defined on page 112. w(n) is defined on page 111.
Transcribed Image Text:9. The Liouville Lambda (A) function is a number theoretic function – similar to T,0,¢ et al. It is denaic by if n = 1 X(12) : (-1)kı+k2+...+kr if n >1 where n = p p...pr e N. = Pi P2 Prove that if n is a perfect square, then X(n) = 1. 10. Let r be a primitive root of some n 2 3. Prove that r ¢(n) = -1 (mod n). 11. Find 7(n), o(n), X(n), µ(n), w(n), and ø(n) for the following integers.' • 2250 • 199 • 286936650 • 22! 12. Let p = 17 and d be a divisor of o(p). Determine (d) for each d. List each element having order d, for all divisors, d, of (p). %3D 13. Calculate all the primitive roots of 41 and 26. 14. Demonstrate that 21 has no primitive root. 15. Let r be a primitive root of n. If gcd(a, n) = 1, then the smallest positive integer k such that a = rk (mod n) is called the index of a relative to r, denoted by ind,a. The theory of indices can be used to solve congruences. Consider the properties of indices (p. 164) and example 8.4 (p. 164). Solve %3D 8x = 11 (mod 13) using the fact that 6 is a primitive root of 13. Construct a table of indices for 6 modulo 13. 16. (7 points (bonus)) Solve the exercise that was previously skipped (since only 14 are required). Be sure to clearly mark the exercise you wish to count for extra credit. Here is an excellent online large number calculator that has a built-in modulo n feature: https://www.calculator.net/big-number-calculator.html 'µ(n) is defined on page 112. w(n) is defined on page 111.
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