Given that a is a quadratic reisdue of the odd prime p, prove the following:                                                                    b) The integer p - a is a quadratic residue or nonresidue of p according as p=1 (mod 4) or p=3 (mod 4).                      c) If p = 3 (mod 4) then x ± a^((P+1)/4)) (mod p) are the solutions of the congruence x^(2) = a (mod p).

Q: let x  be a fixed but arbitrary integer. Do there exist infinity many integer y such that,…

A: Let x = x0 be fixed given integer. Then,

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: 4. Solve each of these congruences using the modular inverses found in the previous problem. (a) 7x…

A: The given congruence equations are (a)    7x=2 (mod 26) (b)   19x=4  (mod 141)

Q: Solve the congruence 11x = 45 (mod 97) where x is an integer in the interval [0,97).

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Q: The polynomial f(x) = x³ − x−3_mod (47) has x₁ = 10 as a root. Use Hensel's Lemma to lift this root…

A: Given Information:The polynomial  is given as .The one of the roots is given as .To find:The lifted…

Q: 10. (8 points) Solve 3x 2, (mod 7) or show that no solution exists.

A: Solution: The objective is to find the solutions of 3x≡2 (mod 7)

Q: 6. (i). Prove that every prime > 3 is of the form 6m +1 or 6m + 5.

A: As per our guidelines we are supposed to solve only one question. For the solution of the first…

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: a) For an odd prime p, prove that the quadratic residues of p are congruent mod p to the integers…

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Q: solve the quadratic congruence 7x^2 - 8x +4 = 0 (mod 17)

A: The given problem is to solve the given quadratic congruence equation 7x2-8x+4 = 0(mod 17)

Q: (b) By applying part (a), solve the congruences 2x = 1 (mod 31), 6x = 5 (mod 11) 3x = 17 (mod 29).…

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Q: Exercise 2.8. Let a, b be real numbers and let a ^= (₁ 1) A and B -(1) What is AB ? Compute A² and…

A: We use method of induction to prove A^m .

Q: Use Hensel’s Lemma and Chinese Remainder Theorem to find all solutions to x^2 + 3x + 17 ≡0 (mod 300)

A: Introduction: Hensel’s Lemma is a powerful tool in number theory that allows us to lift solutions of…

Q: Find the first few terms of the sequence of pseudorandom numbers generated using the linear…

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Q: Find the first few terms of the sequence of pseudorandom numbers generated using the linear…

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Q: Solve the quadratic congruence x² = 35 (mod 817).

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Q: Compute all least residue solutions modulo 12 of 8x-4 congruent to 0(mod 12)

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Q: Need help with True or False for the following... (I will put what I believe As (T) or (F)) 1. For…

A: Let's evaluate the provided statements and their negations as well as the converses and…

Q: If n 2 1, then the Fermat number Fn 22" +1 is prime if and only if %3D 3(F.-1)/2 = -1 mod (Fn).

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Q: Work the modular arithmetic problem. (12 9)(mod 3). (12 9)(mod 3) =(mod 3)

A: The given modular arithmetic problem is -  (12·9)(mod 3) We have to solve the above problem.