Question

Let m > 1. If there exists a primitive root modulo m, prove that there are exactly φ(φ(m)) primitive roots modulo m.

*Note that φ() is Euler's totient function.

Answer #1

Let p be a prime and let a be a primitive root
modulo p. Show that if gcd (k, p-1) = 1, then b≡ak (mod
p) is also a primitive root modulo p.

Given that 2 is a primitive root modulo 19, find all the
primitive roots modulo, 19. You must know how you are getting your
answer and make sure all your answers are in the canonical residue
set

(i) Verify that 2 is a primitive root modulo 29.
(ii) Find all the primitive roots modulo 29. Explain how you
know you have found them all.
(iii) Find all the incongruent solutions to x6 ≡
5(mod 29).

(a) Prove that if y = 4k for k ≥ 1, then there exists a
primitive Pythagorean triple (x, y, z) containing y.
(b) Prove that if x = 2k+1 is any odd positive integer greater
than 1, then there exists a primitive Pythagorean triple (x, y, z)
containing x.
(c) Find primitive Pythagorean triples (x, y, z) for each of z =
25, 65, 85. Then show that there is no primitive Pythagorean triple
(x, y, z) with z...

using discrete logarithms for modulo 17 relative to primitive
root 3,
solve the following:
x^12 (is equivalent to) 13(mod 17)
PLEASE SHOW ALL STEPS IN DETAIL
ANd show ALLL POSSIBLE ANSWERS (values of x)

Use the fact that each prime possesses a primitive root to prove
Wilson’s theorem:
If p is a prime, then (p−1)! ≡ −1 (mod p).

Let p be an odd prime.
Prove that −1 is a quadratic residue modulo p if p ≡ 1 (mod 4),
and −1 is a quadratic nonresidue modulo p if p ≡ 3 (mod 4).

Let f : [1, 2] → [1, 2] be a continuous function. Prove that
there exists a point c ∈ [1, 2] such that f(c) = c.

Find a square root of −1 modulo p for each of the primes p = 17
and p = 29. Does −1 have a square root modulo 19? Why or why
not?

Number Theory:
Find a square root of −1 modulo p for each of the primes p = 17
and p = 29. Does −1 have a square root modulo 19? Why or why
not?

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