Prove that there are no primitive roots modulo 954271 = 691 · 1381 (691 and 1381 are prime). Do this directly (i.e. without referring to the Primitive Root Theorem or any related results).

Q: 4965 4 mod 5 4965 = 2 mod 7 4965 = 1 mod 11 Use your answers above to find the canonical…

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Q: Define prime and semiprime ideals.

A: Topic:- algebra

Q: Suppose that p, q are distinct prime numbers, prove that p9-1 + gp-1 = 1 mod pq

A: We use Fermat's Little Theorem, to prove our claim.

Q: 7) Show that there are no primitive roots modulo 21 using EGLF and CRT.

A: Given value of n is 21 . Now, break 21 into its factor, 21=7×3 ,7,3=1 If m>2 and n>2 then the…

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: Find an integer x satisfying the equation: 4x21 = 2(mod 5) or show that no such integer exists.

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Q: Use the Chinese Remainder Theorem to find a solution to the congruences ´x = 2 mod 10 x = 3 mod 1001…

A: The given system is, x≡2mod10x≡3mod1001x≡3mod9 Chinese reminder theorem: If m1,m2,…,mn are positive…

Q: Let р be a prime such that p = 22" + 1, for some n € N, with n > 1. (a) Determine if 5 is a…

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Q: 4. Find the first three nonzero terms in the Mac expansions of cos(x)/(1+x) by any algebraic method…

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Q: Let p be an odd prime number and an integer ga primitive root with respect to mod p. So is the…

A: We will use basic knowledge of elementary number theory to solve this question.

Q: Describe congruence conditions on p > 2 an odd prime that classify exactly when 11 is a quadratic…

A: We have to describe congruence condition on p>2 an odd prime that classify exactly when 11 is a…

Q: Prove that there are infinitely many primes.

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Q: Prove, that if 2m + 1 is a prime number, then m must be a power of 2 (m = 2n, for some n ∈ N). Prime…

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Q: Recall that tn denotes the nth triangular number. Prove the following identity: tn+2k = tn (mod k)

A: If tn is the nth triangular number, then tn=nn+12

Q: Find the primitive roots (mod 17).

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

A: Quadractic residue: Let p  be any odd prime number and gcd(a,p)=1. Then a is said to be a quadratic…

Q: primes

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Q: Let r be a primitive root of the prime p with p = 1 (mod 4). Show that -r is also a primitive root…

A: Solution

Q: 5 is a quadratic non- residue mod 17 True False

A: 5 is a quadratic non- residue mod 17

Q: Find all the primitive roots modulo 125.

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Q: Find the remainder using Fermat’s Little theorem 452 mod 11.

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Q: Prove that all generators of Z20 are prime numbers. Please provide complete solutions. Thank you.

A: The statement is incorrect as 9 is generator of Z20, but 9 is not prime number.

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

A: We have to state whether the statements are true or false

Q: 5. Use the method of successive squaring to compute 3875 (mod 103) 1

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Q: Suppose that p≥ 5 is prime. Show that -3 is a quadratic residue (mo p) if p = 1 or 7 (mod 12) and a…

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Q: 5. Verify that the prime number 101 divides 52671234505- 57 using Fermat's theorem.

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Q: 1. Give the ten 10th roots of unity. Write them in the form e. Among these 1Oth roots, which are the…

A: According to Bartleby's guidelines, we are allowed to answer only one question at a time. So here is…

Q: find the integer x such that 0 < x < 232 and 144x = 4(mod 233). Show your work.

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Q: a) Show that 2340 ≡ 1 (mod 11) by Fermat’s little theorem and noting that 2340 = (210)34.b) Showthat…

A: According to the given information, Fermat’s little theorem says that:

Q: Use Fermat's Little Theorem to find (Show work.) a) 945 mod.23. b) 231002 mod. 41 c) (-9)933 mod. 13

A: Fermat's Little theorem states that " if p is a prime number and a is not divisible by p then…

Q: Use Hensel's Lemma to find a solution to the following congruence: a +3 x + 3 = 0 mod 54.

A: Let f(x) be a polynomial , p is a prime number . If x=a is a solution of congruence equation…

Q: Number theory

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Q: Find the modular residue of the product. 9,512,667 2,183,321 (mod 5) 9,512,667 2,183,321 (mod 5) =…

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Q: It is given that 3023 is prime, and 3023 = 2(1511) +1 where 1511 is also prime. (a) How many…

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Q: Let odd prime number P =4 (mod L), be an ond If the integer g iso primitive root with respect to mod…

A: Let r be a primitive root of the prime p with p ≡ 1 (mod 4). We have to show that −r is also a…

Q: Prove that every prime greater than 3 can be written in the form 6n + 1 or 6n + 5.

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Q: Let a be a quadratic residue of the odd prime p. (a) Show that a is not a primitive root modulo p.…

A: An integer a is referred to as quadratic residue modulo p if there exists n∈ℤ such that n2≡amod p.…

Q: Use Fermat's theorem to show that for any two distinct odd primes P.9, and any integer C: CM-c-c +c…

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Q: Let n = pq be the product of two odd primes. Recall that there are no primitive roots modulo n. Show…

A: We use the idea that if primitive root exists then x^2 = 1 mod (m) has only 2 solutions. The…

Q: Find (with proof) all positive primes p such that 217p is a Carmichael number.

A: Carmichael number - A number n is a Carmichael number if and only if n=p1p2…pj , a product of at…

Q: 3. Find the residue of 20211202 mod2022 using Euler's Theorem.

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Q: 8. Show that 18!= -1 (mod 437). 9. Using Euler's Theorem, find the last two digits of 44443333 10.…

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Q: 5 is a quadratic non- residue mod 17

A: The quadratic residues mod17 are { 1,2,4,8,9,13,15,16}

Q: Prove that 1 + sqrt5 is irreducible and not prime in z [sqrt5]

A: I have used the definition of prime amd irreducible elements in ring

Q: Find, with proof, all primitive roots mod 27.

A: To find all primitive roots mod 27,

Q: Find the least non negative residue of (194+(13!)44)mod 23

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Q: Use Fermat’s little theorem to find 7121 mod 13.

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Q: Prove that if p is a prime in Z that can be written in the form a2 + b2,then a + bi is irreducible…

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