a) Find a mod p primitive root for p = 19, with the help of this primitive %3D root, make the indèx table.

Q: 3. Prove that for each natural number n, 十2 ...

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Q: 7) Let p be an odd prime, find if a) p =1 mod 8

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Q: For congruences of primes p = 1 (mod 4), a. If a is a quadratic residue modulo p, and p = 1 (mod 4),…

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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: Find the least residue of 63^9 modulo 10

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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: Let r be a primitive root of some n > 3. Prove that r = -1 (mod n).

A: Solution: Let n be any positive integer. Let Un= r : r is less than n and prime to n , where r=r+kn…

Q: Show that: Ifp is odd prime of the form 4k+1, then find a solution of x +1 0 mod p

A: We shall use Euler’s Criterion here. Euler’s criterion states as follows: “Let p be a prime. The…

Q: 2. (- 5 i 3-2i,

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Q: Find the primitive roots (mod 17).

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Q: a) Let p be an odd prime. Show that a = -1 mod p has a solution if and only if p = 1 mod 8.

A: Consider the provided question,Hello. Since your question has multiple parts, we will solve first…

Q: Hello, I need help for the problem in the picture. Thank you!

A: We are given that α ∈ Z is a primitive root modulo an odd prime p. So,    αϕ(p) ≡ 1 mod p => αp-1…

Q: Prove that 2" >n For all nuatural numbers

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Q: Find a primitive root mod 172021

A: If a is a quadratic residue mod 17, then a8≡1 mod 17. Thus order of a≤8. Hence a cannot be a…

Q: 4. Let p be an odd prime, and a, b primitive roots mod p. Show that ab is not a primitive root mod…

A: It is given that, p be an odd prime, and a and b primitive roots mod p.

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: The number p = 65537 = 210 +1 is known to be prime; you may take this as given. Let g be a primitive…

A: Since, g is a primitive root modulo p, order of g is (p-1) = 216 Now, we know order of gk is…

Q: Show that every nonzero residue (mod 17) can be written as a power of 3

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Q: Suppose g is a primitive root mod p (with p an odd prime) such that g +1 mod p? Prove g must be a…

A:

Q: Show that if p is an odd prime, then n = 2º – 1 passes Miller's test base 2 -

A: Given that n=2p-1 where, pis odd prime. Since, 2p is an even integer. Then, n=2p-1 is an odd…

Q: Find all the primitive roots modulo 125.

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Q: Show that the modulus of the quotient of tow complex conjugate numbers is 1, that is :1.

A: Use the following formulae, to prove the required result. The conjugate of the complex number a+ib…

Q: Let p be a prime and a a quadratic residue of p. Show that if p =1 (mod 4), ais also a quadratic…

A: Let P be an odd prime and Let a be a quadratic residue mod p ∵ P-a≡-a (mod p) ∴(P-a)p-12≡(-a)p-12…

Q: 5. Let p be an odd prime, and suppose that ged(a, p) = 1. Show that the congruence z = a (mod p*)…

A: p is an odd prime and gcd(a,p)=1We have to show the congruence x2=a mod pk has exactly 1+ap…

Q: Let m e N and a E Z. Suppose that 3|(m) and that a is a primitive root (mod m). Prove that a is not…

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Q: Find the smallest positive integer n, пп such that 0 and 0 are 2 coterminal. And give an example

A: Coterminal angles are the angles in the standard position having a common terminal sides.

Q: Let p be an odd prime, and let g be a primitive root modulo p. Prove that -g is a primitive root…

A: Given : p= odd prime g= primitive root modulo p

Q: Suppose that residue(modp) 8) Let odd prime P. atzis Show that has be an not Diophantine…

A: Given that a is not a quadratic residue mod p. That is there does not exists an integer z such that,…

Q: Let be a rational prime, with p = 1(mod 4). Prove that p is the product of two Gaussian primes that…

A: We will use the basic knowledge of number theory to answer this question properly and completely.

Q: 8. Use Euler's Theorem to find a number a between 0 and 9 such that a is congruent to 7 1000 modulo…

A: We know according to  Euler's Theorem  that  xϕ(n)≡1(mod n) where the function ϕ(n) counts the…

Q: Find a primitive root modulo 2 * 19^2022

A: Consider the number n=2×192022. We know that a primitive root of modulo n is an integer g such that…

Q: Problem 2. Let p be an odd prime number and let r be a primitive root (mod p). Thus, płr, rP-1 = 1…

A: Let p be an odd prime and r be a primitive root modulo p. ... Can Write A = R^k For Some K.  Prove…

Q: Find all primes p < 500 such that there are exactly 18 elements that have order 19 modulo p.

A: Given that p is prime . Let us take p=19 By Fermat's little theorem, " If p is a prime and a is any…

Q: find a primitive root mod 172021

A: to find the primitive root of 172021 definition of primitive root let n be a prime integer then g is…

Q: (a) Let p be an odd prime. Give an analogue of the quadratic formula mod p, and prove it. (b)…

A: (a) Given that p is an odd prime.  The quadratic formula mod p is given by: ax2+bx+c≡0mod p Here, p…

Q: Use Dirichlet's theorem to show that there are infinitely many primes whose dicimal expansion ends…

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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: Show the p in th

A: Vertices of the triangle is Given integral is

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: Find a such that 5= 7* modulo 13.

A: Given: 5≡7xmodulo13. The objective is to find the value of x.

Q: a. For each odd prime p, a residue a in Z*p cannot be both a primitive root and a quadratic residue.…

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Q: If a is a primitive root modulo n, then a¯1 is a primitive root modulo n O False True

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Q: Find the least residue of 99! (Mod 101)

A: Introduction: In number theory, Wilson's theorem is a very significant theorem. It states that for…

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: The given problem is to show that -r is the primitive root of p when r is a primitive root of prime…

Q: 3. Find all primes p < 500 such that there are exactly 18 elements that have order 19 modulo p.

A: Step:-1 Since, we know for a prime p, modulo p forms a cyclic group. And, also according to a known…

Q: To index Qn [6, 20] using N as the index set.

A: We need to find ℚ∩6, 20 and we have the index set is ℕ Now, ℚ∩6, 20=6, 20

Q: Let k be an integer such that k = -1 (mod 5). What is the remainder when 50k10 + 26k³ – 17 is…

A: Since you have asked multiple questions, we will solve the first Question  for you. If youwant any…

Q: What is the modulus and argument of 3i.

A: Let z be a complex number then, z is written as z=x+ιywhere x and y are real numbers.We have to find…