Suppose p is a prime congruent to 3 modulo 4. (a) Prove that the congruence r = -1 (mod p) has no solution Hint: Use Fermat's Little Theorem. (b) Show that there are no clements of order 4 modulo p.

Q: Let p be a prime. Show that -2 is a quadratic residue modulo p if and only if p = 1 mod 8 or p = 3…

A: Definition(Quadratic Residue Modulo): In number theory an integer qis called a quadratic residue…

Q: (10) Find all solutions of the congruence 2 = 22 (mod 53). C

A: We shall be using primitive roots to solve this congruence.

Q: Show that 2 is a primitive root modulo 3" for all n > 1.

A: To prove that 2 is a primitive root modulo 3n for all n≥1.

Q: 1) Compute the order of a modulo 7 for a = 1,2,....6. Find all primitive roots modulo 7.

A: Order of the element

Q: Prove the following: (a) For an odd prime p, the quadratic residues of p are congruent modulo p to…

A: Answer :

Q: Show that if p is an odd prime and a and b are integers with a ≡ b (mod p), then

A: According to the given information, let p be an odd prime and a and b are integers with a = b…

Q: 2. If a = 3 (mod 11) and b = 9 (mod 11), compute the values, positive and fully-reduced modulo the…

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Q: (7) (Z/50Z)*. Which of the other elements of (Z/50Z)* are also primitve roots modulo 50? Find a…

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Q: Suppose that a is an odd integer and (a, 91) = 1. Prove that a12 ≡ 1 (mod 1456).

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Q: 8) Prove that 2 is a primitive root modulo 53.

A: We have to Prove that 2 is a primitive root modulo 53.

Q: Suppose p is an odd prime number, prove that p+1 (р — 3)! %3D (mod p) -

A: To prove: To prove that p-3!≡-p+12(mod p) for an odd prime number p.

Q: Prove that if x2 is congruent to y2 (mod n) and x is not congruent +or - y (mod n)  then gcd(x + y,…

A: Given: x2≡y2modn x≢±ymodn

Q: Find all x EZ221 such that x2 =97 (mod 221).

A: To find : all x∈ℤ221 such that x2≡97mod 221 Pre-requisite : Properties of Legendre symbol - 1. ap=1,…

Q: Show that if b0 = 0, then (g^xm)≡ 1 (mod p). Q2: Show that if b0 = 1, then (g^xm) ≡ p − 1 (mod p).

A: Given: x is a number in the set {0, 1, . . . , 2t} x written as: x = b0 · 20 + b1 · 21 + b2 · 22 + ·…

Q: Ulia primeive Toute 95. Find all primes p ≤ 29 such that 2 is a primitive root modulo p.

A: Here we have to find all primes less than 29 or equal to 29 having primitive root 2.

Q: Second Assignment: O Use Euler's Theorem to establish the following: 37 = a mod 1729. [Hint: 1729 =…

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Q: 1. Let a and b be natural numbers, where a, b > 2, such that gcd(a, b) = 1 and ab = (x + y)" for…

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Q: Compute 7^35 (mod 3) ..

A: As per our guidelines, we are supposed to solve only first subpart. Kindly repost other subpart as a…

Q: Prove that  (5/p)=1 if and only if p=1, 9, 11,19 (mod 20)

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Q: Consider congruence mod 5. Let r = {0, 1, 2, 3, 4}. 1) Find r so that 3 =r mod (5). 2) Find r so…

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