Question

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).

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.

In number theory, Wilson’s theorem states that a natural number
n > 1 is prime
if and only if (n − 1)! ≡ −1 (mod n).
(a) Check that 5 is a prime number using Wilson’s theorem.
(b) Let n and m be natural numbers such that m divides n. Prove the
following statement
“For any integer a, if a ≡ −1 (mod n), then a ≡ −1 (mod m).”
You may need this fact in doing (c).
(c) The...

Let p be an odd prime, and let x = [(p−1)/2]!. Prove that x^2 ≡
(−1)^(p+1)/2 (mod p).
(You will need Wilson’s theorem, (p−1)! ≡−1 (mod p).) This gives
another proof that if p ≡ 1 (mod 4), then x^2 ≡ −1 (mod p) has a
solution.

Let p be be prime and p ≡ 1 (mod 4|a|). Prove that a is a
quadratic residue mod p.

Prove: If p is prime and p ≡ 7 (mod 8), then p |
2(p−1)/2 − 1. (Hint: Use the Legendre symbol (2/p) and
Euler's criterion.)

Please solve in full detail! Use the fact that
Zp*, the nonzero residue classes modulo a
prime p, is a group under multiplication to establish Wilson’s
Theorem.

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 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.

can we use fermat's little theorem to prove a number
is prime?

Use the intermediate value theorem to prove that the
equation
ln? = ? − square root(?) has atleast one solution between ?=2
and ?=3

ADVERTISEMENT

Get Answers For Free

Most questions answered within 1 hours.

ADVERTISEMENT

asked 51 seconds ago

asked 3 minutes ago

asked 3 minutes ago

asked 16 minutes ago

asked 16 minutes ago

asked 17 minutes ago

asked 17 minutes ago

asked 36 minutes ago

asked 37 minutes ago

asked 50 minutes ago

asked 50 minutes ago

asked 54 minutes ago