Find a primitive root for: (a) n = 18, (b) n = 50 (c) n =...

Find a primitive root for: (a) n = 18, (b) n = 50 (c) n = 27, (d) n = 625.

Expert Solution

Colby Messinger answered 1 year ago

(a) Let n be odd and ω a primitive nth root of 1 (means that its...

(a) Let n be odd and ω a primitive nth root of 1 (means that its order is n). Show this implies that −ω is a primitive 2nth root of 1. Prove the converse: Let n be odd and ω a primitive 2nth root of 1. Show −ω is a primitive nth root of 1. (b) Recall that the nth cyclotomic polynomial is deﬁned as Φn(x) = Y gcd(k,n)=1 (x−ωk) where k ranges over 1,...,n−1 and ωk = e2πik/n is...

7. (16 pts) a. Show that 11 is a primitive root of 13. b. What is...

7. (16 pts) a. Show that 11 is a primitive root of 13. b. What is the discrete logarithm of 4 base 11 (with prime modulus 13)?

probability 10%. n= 50 find between 15 and 18 times.

Find a b c of a right triangle. with a perimeter of 18.

Find a primitive root modulo 2401 = 7^4. Be sure to mention which exponentiations you checked...

Find a primitive root modulo 2401 = 7^4. Be sure to mention which exponentiations you checked to prove that your ﬁnal answer is indeed a primitive root. (You may use Wolfram Alpha for exponentiations modulo 2401, but you may not use any of Wolfram Alpha’s more powerful functions.)

Find a primitive root, for all positive integral m, modulo each integer below. (a) 7m (Hint:...

Find a primitive root, for all positive integral m, modulo each integer below. (a) 7m (Hint: Using Corollary 5.15, find a common primitive root r modulo 7 and 72• The proof of Proposition 5.17 then guarantees that r is a primitive root modulo 72 for all positive integral m.) (b) 11m (c) 13m (d) 17m

Let p be an odd prime. (a) (*) Prove that there is a primitive root modulo...

Let p be an odd prime. (a) (*) Prove that there is a primitive root modulo p2 . (Hint: Use that if a, b have orders n, m, with gcd(n, m) = 1, then ab has order nm.) (b) Prove that for any n, there is a primitive root modulo pn. (c) Explicitly find a primitive root modulo 125. Please do all parts. Thank you in advance

An element a in a field F is called a primitive nth root of unity if...

An element a in a field F is called a primitive nth root of unity if n is the smallest positive integer such that an=1. For example, i is a primitive 4th root of unity in C, whereas -1 is not a primitive 4th root of unity (even though (-1)4=1). (a) Find all primitive 4th roots of unity in F5 (b) Find all primitive 3rd roots of unity in F7 (c) Find all primitive 6th roots of unity in F7...

(a) find the margin of error for the values of c, s, and n, and (b)...

(a) find the margin of error for the values of c, s, and n, and (b) construct the confidence interval for m using the t-distribution. Assume the population is normally distributed. 16. c = 0.99, s = 16.5, n = 20, x = 25.2

(a) find the margin of error for the values of c, s, and n, and (b)...

(a) find the margin of error for the values of c, s, and n, and (b) construct the confidence interval for m using the t-distribution. Assume the population is normally distributed. 16. c = 0.99, s = 16.5, n = 20, x = 25.2

Question

Find a primitive root for: (a) n = 18, (b) n = 50 (c) n =...

Solutions

Expert Solution

Related Solutions

(a) Let n be odd and ω a primitive nth root of 1 (means that its...

7. (16 pts) a. Show that 11 is a primitive root of 13. b. What is...

probability 10%. n= 50 find between 15 and 18 times.

Find a b c of a right triangle. with a perimeter of 18.

Find a primitive root modulo 2401 = 7^4. Be sure to mention which exponentiations you checked...

Find a primitive root, for all positive integral m, modulo each integer below. (a) 7m (Hint:...

Let p be an odd prime. (a) (*) Prove that there is a primitive root modulo...

An element a in a field F is called a primitive nth root of unity if...

(a) find the margin of error for the values of c, s, and n, and (b)...

(a) find the margin of error for the values of c, s, and n, and (b)...