a. For each odd prime p, a residue a in Z*p cannot be both a primitive root and a quadratic residue. b.. The integer 2 has a quadratic residue modulo the prime 399989.

Q: m and n be positive integers so that m n. Show for any integer a that the congruence

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Q: 10. Prove that if o is the m – cycle (a,a2 ... am), then for all i e {1,2, ...m}, o'(ar) = ar+i,…

A: Given that σ is the m- cycle (a1,a2,...am). We need to prove (i) σi(ak)=ak+i mod m for all i ∈{1, 2,…

Q: a) Prove that if p is prime and p  1≡ (mod 4), then ((p-1)/2)!)^2 ≡ -1 (mod p) b) Use part a) to…

A: According to the given information in part (a) it is required to prove that:

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: 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: For each of the following primes, find the smallest positive integer k such that k is a prim root…

A: A primitive root of a prime p is an integer g such that g(modp) has multiplicative order p-1.

Q: Suppose a is an integer that is relatively prime to 400. Prove that a40 = 1 (mod 400).

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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: Show that if p is an odd prime, then every divisor of the Mersenne number 2p − 1 is of the form 2kp…

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Q: 1. Find a reduced residue system modulo each of the following integers. a) 6 b) 9 c) 10 d) 14 e) 16…

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Q: If p is a prime with p=3 mod 4 and a is quadratic residue mod p, then -a is a quadratic residue mod…

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

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Q: 1. Let n = pq, where p and q are distinct odd primes. Define X(n)= (p-1)(q-1) gcd(p-1,q-1)* Suppose…

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Q: If p > 2 is a prime of the form 2m +1, prove that m is a power of 2.

A: Let p be a prime and p > 2. And, p = 2m +1 Suppose 't' is an odd prime factor of m. That is, m =…

Q: Let be a prime such that p = 1 (mod 4). Prove that –1F, is a square in Fp. -

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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: 2. Find the least nonnegative residue modulo m for each integer n below: (a) n= 100!, m= 101 (b) n =…

A: We have to find the least nonnegative residue modulo m for each integer n below : (a) n=100!, m=101…

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: 3. Use the above theorem to prove that if p E N is a prime number such that p = 1 mod 4, then p is…

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Q: Let o (n) be the sum of the divisors of n and let (n) be the Euler phi functions. Let n = = pq be…

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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 a be a quadratic residue of the odd prime p. (1) Prove that a is not a primitive root modulo p…

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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: Find all the positive primes p such that 9p+ 1 is a perfect cube. Namely, such that there exists an…

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Q: Suppose that p is a prime such that p = 5 (mod 8). Let k E Z such that p = 8k+5. For any a E Zp,…

A: 1. Let a∈ℝn. Then a is called quadratic residue modulo n if there exists an integer x such that…

Q: 10. Let n be an even perfect number. Prove that n=2*(2k+1 – 1), for some integer k.

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Q: If p is a prime and k is any integer, then S = {1*, 2*, 3*, .., (p - 1)*} is a reduced residue…

A: A subset S of integers is called a reduced residue system modulo p if : (i) gcd(a,p)=1 for each a∈S…

Q: brvib - If m is composite, what is the value (m – 1)! mod m in the standard residue system? Conclude…

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Q: Find all positive primes p such that n = 1001p satisfies that a" = a mod n for all a.

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Q: A sequence {an} is defined recursively by a1 = 6, a2 = 8, and an = 2an−1 − an−2 for n ≥ 3. Prove…

A: A sequence {an} is defined recursively by a1 = 6, a2 = 8, andan = 2an−1 − an−2 for n ≥ 3.  To Prove…

Q: Prove for each integer n, that n2 is congruent either to 0 or to 1 modulo 4.

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Q: 1. Use Euler's Theorem to prove a 265 = a (mod 105) for all a E Z.

A: To prove:                   a265≡amod 105 for all a∈ℤ by using Euler's theorem.   Euler's theorem:…

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: Let a, n E Z+, where gcd(a, n) = 1. Show that if n is a pseudoprime to the base a, then n is a…

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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: Suppose that n > 2 is an even integer and a is an (n – 1) -cycle in S,. Show that a does not commute…

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Q: Consider numbers of the form 9" mod 13, where n E Z+. We claim that all of these numbers will only…

A: First principle of mathematical induction : Suppose Pn is a statement where n∈ℤ+ . The statement is…

Q: Show that if p is an odd prime, then −1 is a quadratic residue of p if p ≡ 1 (mod 4), and −1 is not…

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Q: Let p be a prime such that p mod 4 = 3. How many elements oforder 4 are in GF(pn)*?

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Q: Let p be a prime, a ∈ Z. Prove that ap − a is divisible by p.

A: Fermat's Theorem : If p is any prime and a is any integer then p divides (ap-1-1).

Q: 4) The least positive integer residue of 21000000 mod 17 is a) - 1 b) 1 c) 2 d) a

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Q: Let p be a prime such that p = 1 (mod 4). Prove that -lF, is a square in Fp.

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Q: Let be an odd prime. Prove that for all i with 1 sisp-2 one has p| 1° + 2' + ... + (p – 1)'.

A: Given : p is an odd prime To Prove :          p| 1i+2i+. . .+p-1i for all i with 1≤i≤p-2

Q: 1. It has been conjectured that there are infinitely many primes of the form n2 – 2. Exhibit five…

A: See full solution below

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: 2.1.3. If a, b are integers such that a = b (mod p) for every positive prime p, prove that a = b.

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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: Let n > 2 be a positive integer and r a primitive root of n. Suppose that a E Z is such that ar? 1…

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