Assune p is an odd prime such that p = 1mod4 (that is p 4k + 1) Prove: if a is a primitive root for p, then so is -a (so, we're assuming ord,(a) = p- 1 and we have to show ord,(-a) p-1, by checking the two conditions of order)

Q: Let p= 2k +1 be an odd prime number. Show that x* – 1 = (X – 1)(X – 2²) - · (X – k²) mod p.

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Q: If 2n +1 is prime for some n ≥ 1, prove that n is a power of 2.(Primes of the form 2n +1 are called…

A: Given: 2n+1 is prime for some n≥1. To Prove: n is a power of 2.

Q: 3. Let a be an odd integer. Prove by induction that a² - 1 is divisible by 2+1 for every integer n ≥…

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Q: Prove that the only prime of the form n² - 4 is 5.

A: Suppose p is a prime such that p=n^2-4. Then we have p=n^2-4=(n+2)(n-2). we know that prime number…

Q: Work the modular arithmetic problem. (9.7)(mod 5). (9.7)(mod 5) = (mod 5)

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Q: Prove that if p and q are distinct primes, then pq | (p9-1+qp-1-1).

A: p and q are distinct prime. We have to show that pq|pq-1+qp-1-1

Q: For integer n>1, nZ={nx:x in z}, nZ is set of all integer multiples of n. Prove that integer p is…

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Q: (a) Show that -163 37 Actually, the following more general result is true: -163 р -163 31 = -1. = -1…

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

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Q: Given n > 1, let p be a prime number such that n < p < 2n (by the way, 2n there is always a prime…

A: here we need to expand  by using  nr=n!r!×n-r! and then show that expression obtained have a factor…

Q: a)Consider the following data: 3 is a quadratic residue for 23 and 37. 3 is a quadratic nonresidue…

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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: Prove A prime is irreducible but not conversely.

A: suppose that a is a prime in an integral domain and a=bc. we must show that b or c is a unit. by the…

Q: if p. (144) =(-¹) ove tha pand gare distin (p-1)(g-1) 4 aci

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Q: Let p be a prime number. Prove that (p -1)! a power p + (3p power2  +1)a is divisible by p, for any…

A: Consider the provided question, (a)

Q: ak written in the form n = pi p²p with a 20 and p, are distinct prime. The factoriz if we order the…

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Q: 3. Exercise 55.2 #12. Let p be an od«

A: Given, p be an odd prime number.

Q: Fill in the blanks in the proof of the of fact that 3" – 1 is a multiple of 2 for every natural n.…

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Q: Suppose |G| = p2q2 , where p < q are primes. Prove that if |G| ̸= 36, then nq = 1.

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Q: The Fundamental Theorem of Arithmetic states that if n 1 is an integer then e1e2 n = = P₁¹p²/² ek…

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Q: c.) There is a conjecture called "Goldbach's weak conjecture" (yes, really!) that every odd number…

A: Given: Odd numbers greater than 5. To prove: Every odd number greater than 5 can be written sum of…

Q: Q4 (a) Solve for x if 25x = 15(mod 29). (b) Determine the smallest positive integer so that when…

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Q: Prove: Let be odd prime. Prove that Aor all i with 1 LikP-2), you have an +... + working so far:…

A: As per the question we just have to show that the prime p divides the following : xi + x2i + ... +…

Q: Find the canonical prime factorization of this using Fermat’s Method and solve for the Number of…

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Q: 7 .Assune p is an odd prime such that p ≡ 1mod4 (that is p = 4k+1) Prove: if a is a primitive root…

A: Given: p≡1mod4       ∵p=4k+1

Q: (b) Find a prime p> 5 such that 2 and 3 are both primitive roots modulo P.

A: We test all the primes p&gt;5 and see which prime works.

Q: , Let p be an odd prime. Prove or disprove the following statements. (a) If p = 2 + 1, then every…

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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: . (a) How many quadratic residues are in the residue set Z19? (b) Determine if 4 and 9 are quadratic…

A: Introduction: Quadratic residue is related to congruent modulo theory. Since, it deals with…

Q: Work the modular arithmetic problem. (8.9)(mod 4). (8.9)(mod 4) = (mod 4)

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