|State whether the following statements are true or false. Show working. a. For each odd prime p, a residue a in Z*p cannot be both a primitive root and a quadratie residue. b. The integer 2 has a quadratic residue modulo the prime 399989. c. If p, q are odd primes and pg = 1 mod 4, then –1 has a quadratic residue modulo at least one of p or q.

Q: Prove that there are infinitely many primes congruent to 3 mod 8. previous part.

A: We have to show that there are infinitely many primes congruent to 3 mod 8. we will proof this by…

Q: Use the Extended Euclidean Algorithm to find d = ged(85, 210) and write d as an integer combination…

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Q: Prove that for all integers n ≥ 1, 10" = (-1)" (mod 11). Use part (a) to prove that a positive…

A: (a) Let n≥1 be an arbitrary integer. We know that 10≡10mod 11≡-1mod 11. Now consider 10n:…

Q: Suppose p is a prime such that p ≡ 1 (mod 4). Show that the sum of the quadratic residues mod p…

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Q: 4444 is divided by a 1.19) find the remainder when 4444 * 2?=-1 (mod a).

A: 1.19 We have 4444  ≡  7 (mod 9). So, 44444444 ≡ 7 4444 (mod 9) Now,       73 ≡1 (mod 9) Or,…

Q: 1. Show that k1601 = k (mod 17) for any k.

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Q: There exists an integer m > 3 such that m2 – 1 is prime. The above statement is due to proof by…

A: We have to complete the given sentence.

Q: 13. Compute 41023 (mod 11). a. 14. Prove that for any integer n, n31 = n(mod7).

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Q: Recall that an integer m is a square if m= k² for some integer k and is a triangular number if m =…

A: This is a classic application of pells equation. We first transform the equation into a pells…

Q: |State whether the following statements are true or false. Show working. a. For each odd prime p, a…

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Q: For which primes p > 5 is -5 a square mod p? Your answer should be of the formp = {a1, a2, ..., ak}…

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Q: For any integer a, prove that a=0 or 1 (mod 3).

A: The solution is given as follows

Q: Theorem 1. Let r, y, z be positive integers where r is odd and y is even. Then r, y, z is a…

A: As per Bartleby guidelines for more than one questions asked only first should be answered. Please…

Q: Work the modular arithmetic problem. (11.10)(mod 7). (11 10) (mod 7) = (mod 7)

A: (a)   (11.10)mod 7  =  110 mod 7                            The remainder when 110 is…

Q: If a = 8(mod19) and b = 6(mod19), which of the following is the least residue of a × b modulo 19? ●…

A:

Q: 5. Using congruence theory (not brute force), find all solutions to the following linear congruence:…

A: Note:- A two variable linear congruence ax+by≡m    mod (n) has solution if and only if  gcd ( a, b,…

Q: . Solve the following modular arithmetic. Show your solution in trhe space provided. 4. ( 30/7 )…

A: If a is congruent to b modulo n then n divides a-b.

Q: Let p E Z Z be a prime number such that p > 2. Let n = 2p. Find all positive %3D divisors of n. NO…

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Q: Find the primitive roots (mod 17).

A: Let p be a prime number and g be a positive integer. Then, g is a primitive root (mod p) if and…

Q: 7. Prove that if n is an odd positive integer or divisible by 4, then 13 +23 + 33 + .+ (n– 1)3 = 0…

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Q: 2. Let p be a prime number such that p> 5. Prove that p2-1 =0 (mod 24).

A: To prove the below statement If p is a prime number and p > 5. p2−1≡0mod24

Q: Select the following statements that are true. 261000 = 1 (mod 3) (2022) 10 (FC6) 16 In Z6, 2¹⁰ = 2…

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Q: Prove that a positive integer a > 1 is a square if and only if in the canonical form of a all the…

A: Since you have posted multiple questions but according to guidelines, we will solve first question…

Q: 1.34. This exercise begins the study of squares and square roots modulo p. (a) Let p be an odd prime…

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Q: Prove or disprove the following proposition: Let the Jacobi symbol: () 1. %3D Then a is quadratic…

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Q: 3. Prove the following two theorems about pairs of "twin primes," p and (Recall that "twin primes"…

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Q: Prove by Contradiction: There are no integers x and y such that = 4y+2.

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Q: 1. Let z = 46200. Write it in the factorized form (1) where ni > 0 and p; are distinct primes, using…

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Q: (a) Characterize the primes p for which 63 is a quadratic residue.

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Q: Find the largest positive integer p for which the R computation 2°p yields a finite representable…

A: Given: Determine the biggest positive integer p for which the R calculation 2^p produces a finitely…

Q: Show that n := e2™i/n is a primitive nth root of unity in C. Verify that (S3 - )? = -3 in C. Verify…

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Q: For each pair of integers m and d, calculate m mod d. Show your work! a. m = 43 and d = 9 b. m 50…

A: If m and d be two integers . Then m mod d means that the remainder when m is divided by d .

Q: Exercise 6.4. We say that a number î divides another number z if there exists an integer k such that…

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Q: Recall the Fibonacci numbers Fn are defined by Fo = 0 , F1 = 1 and Fn Fn-1 + Fn-2 for all n > 2 .…

A: Consider the provided question, Fibonacci numbers Fn are defined by F0=0, F1=1 and Fn= Fn-1+Fn-2 for…