1. (i) Show by hand that 2 is a primitive root mod 19.
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter1: Fundamental Concepts Of Algebra
Section1.2: Exponents And Radicals
Problem 92E
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[Number Theory] How do you solve question 1?
![1. (i) Show by hand that 2 is a primitive root mod 19.
(ii) Find² a complete set of primitive roots mod 19.
2. Suppose that a and b are primitive root modulo an odd prime p. Show that ab is not a
primitive root mod p.
3. Suppose that p and q are odd primes. Show that pq is a pseudoprime base 2 if and only
if ord, (2)| (q-1) and ord, (2)| (p-1). Here, if gcd(a, p) = 1, we use the notation ord, (a)
for the order of the element [a], in the group Up.
4. (i) Show that 2 is a primitive root mod 11² = 121. Hint: Use the criterion given by
Lemma 6.4 and a calculator. Conclude that 2 is a primitive root mod 11e for any e.
(ii) How many elements of order 5 are there in the group U1331? Note that 1331 = 11³.
(iii) Find all elements of order 5 in U1331.
¹1.e., do not use a calculator
2Here you may use a calculator to compute residues mod 19.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6b892631-cf27-4f8a-b2dd-bfb238d2c250%2Fa24cfb74-7f5a-4994-b631-9e39d389db4d%2Fwmorz5r_processed.png&w=3840&q=75)
Transcribed Image Text:1. (i) Show by hand that 2 is a primitive root mod 19.
(ii) Find² a complete set of primitive roots mod 19.
2. Suppose that a and b are primitive root modulo an odd prime p. Show that ab is not a
primitive root mod p.
3. Suppose that p and q are odd primes. Show that pq is a pseudoprime base 2 if and only
if ord, (2)| (q-1) and ord, (2)| (p-1). Here, if gcd(a, p) = 1, we use the notation ord, (a)
for the order of the element [a], in the group Up.
4. (i) Show that 2 is a primitive root mod 11² = 121. Hint: Use the criterion given by
Lemma 6.4 and a calculator. Conclude that 2 is a primitive root mod 11e for any e.
(ii) How many elements of order 5 are there in the group U1331? Note that 1331 = 11³.
(iii) Find all elements of order 5 in U1331.
¹1.e., do not use a calculator
2Here you may use a calculator to compute residues mod 19.
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