1.32. Recall that g is called a primitive root modulo p if the powers of g give all nonzero elements of Fp. (a) For which of the following primes is 2 a primitive root modulo p? (i) p = 7 (ii) p = 13 (iii) p = 19 (iv) p = 23 (b) For which of the (i) p = 5 following primes is 3 a primitive root modulo p? (ii) p = 7 (iii) p = 11 (iv) p = 17
1.32. Recall that g is called a primitive root modulo p if the powers of g give all nonzero elements of Fp. (a) For which of the following primes is 2 a primitive root modulo p? (i) p = 7 (ii) p = 13 (iii) p = 19 (iv) p = 23 (b) For which of the (i) p = 5 following primes is 3 a primitive root modulo p? (ii) p = 7 (iii) p = 11 (iv) p = 17
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
Section: Chapter Questions
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Transcribed Image Text:1.32. Recall that g is called a primitive root modulo p if the powers of g give all
nonzero elements of Fp.
(a) For which of the following primes is 2 a primitive root modulo p?
(i) p = 7
(ii) p = 13
(iii) p = 19
(iv) p = 23
(b) For which of the following primes is 3 a primitive root modulo p?
(i) p = 5
(ii) p = 7
(iii) p = 11
(iv) p = 17
(c) Find a primitive root for each of the following primes.
(i) p = 23
(ii) p = 29
(iii) p = 41
(iv) p = 43
(d) Find all primitive roots modulo 11. Verify that there are exactly (10) of them,
as asserted in Remark 1.33.
(e) Write a computer program to check for primitive roots and use it to find all
primitive roots modulo 229. Verify that there are exactly (229) of them.
54
Exercises
(f) Use your program from (e) to find all primes less than 100 for which 2 is a
primitive root.
(g) Repeat the previous exercise to find all primes less than 100 for which 3 is a
primitive root. Ditto to find the primes for which 4 is a primitive root.
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