Theorem 1. Let c E N be a natural number and 2k .p21 its factorization into distinct primes with pi = 1 mod 4 and qj = 3 mod 4. m m Then #{(a,b) Z2 a2 b = c and a > b>0 if there exists j e [m] such that li 0 = 1 mod 2 П. (ki1) else i=1

Q: 1.2. Problem 2. Let 1 < a < b be two positive integers. Prove or disprove: there exists a…

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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: Let n = pq, where p and q are distinct primes such that p = q = 3 (mod 4). Suppose a E Z+ with a <…

A: We use the properties and definitions of Jacobi symbols to solve this problem. The detailed solution…

Q: Theorem 8. If o is even and k, 1 are odd positive integers and (A+D+1) + (B+C), then Eq.(1) has no…

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Q: Prove by induction that for all n > 2, Ln = Fn-1 + Fn+1 Where La denotes the nth Lucas number and Fn…

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Q: Theorem 1. Every positive integer n > 2 can be written as a product of primes. In other words, for…

A: Either n is a prime or it is composite; in the former case, there is nothing more to prove. If n is…

Q: . Consider a = 7+ 2i andB = 3 – 4i in Z[i]. Find o and p in Z[i] such that ine = βσ+ ρ with N(p) <…

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Q: Give an inductive proof that the Fibonacci numbers Fn and Fn+1 are relatively prime for all n ≥ 0.…

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Q: Make an outline for a proof by induction that 2 = n(n+1)(2n + 1) for all n E Z+. k=1

A: Note: We can solve one question at a time so kindly upload second question as another question.

Q: Let p be a odd prime number and b, c € Z. Assume that there exists l e Z, such that l² = b² – 4c mod…

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Q: and C of S such that oB oc (mod 6). 5.50. Prove that the equation cos (x) – 4x + 7 = 0 has a real…

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Q: Show that a fraction r = a/b in lowest terms has a finite decimal expansion if and only if…

A: Given, r = a/b, where b = 2n5m for some n, m ≥ 0. We know, that any integer divided by 10N has a…

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: (7) In the ring (Z, +, .), we get n {P:P non trival prime ideal in Z}: (a) o (b) (Z, +, .) (c)…

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Q: Show that the last digit of the Fermat number Fn =2^(2^n)  is 7 for all integers n ≥ 2. [Hint: By…

A: We have to prove that the last digit of the Fermat number Fn=22n is 7 for all integers n≥2. For…

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: 1. In the proof of the claim "there is no integer n such that n = 2 mod 6 and n = 7 mod 9," the…

A: See answer below

Q: For p an odd prime in Z , Prove: If -1 is a perfect square in Z/pZ then p= 1(mod 4)

A: This can be done as follows:

Q: If z = cosθ + isinθ,  where -π < θ <= π.  Find the modulus and argument of: (a) z2  (b) 1+ z2 ,…

A: Please rate and feel free to ask any query about any part of the question.  Thanks

Q: 11- P, X Pz X P, ... X Pn, where M= Pr. Pn are distind primes Prove by inducton, For all Ppositive…

A: Fermat's little theorem is one of the most important and significant methods in number theory.…

Q: Problem 1. (1) What are the squares modulo 4? (2) Let p be an odd prime and assume that p= a²+t for…

A: Hello. Since your question has multiple parts, we will solve first question for you. If you want…

Q: [1E q14] Question is attached, Thanks ?

A: From the given information it is needed to prove that:

Q: 5. The Fibonacci numbers are defined as: fo = 0 , fi = 1 , and fn = fn-1 + fn-2 , for n > 2. Give a…

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Q: 2. Find the Eurler's function of 45, i.e., (45) =? 3. For n EN and p is prime, prove that (nlp - 1…

A: According to our guidelines we can answer only one question and rest can be reposted.

Q: Let p be a prime divisor of the Fermat number Fn 22" + 1. : 2n+1 (a) Show that ord, 2 (b) Use…

A: As per the question we have the Fermat number Fn = 22ⁿ+1 and its prime divisor p. Here we will use…

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: Let n E Z', where n > I. Show that if n is a product of distinct Mersenne primes, then o(n) = 2k for…

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Q: Show that if n = p1p2⋯pk, where p1, p2,…, pk are distinct primes that satisfy pj − 1 | n − 1 for j =…

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Q: 9. Let p be an odd prime, and let ( - 1)(a – 2).. (a - (p - 1)) = "--g12-2 -3 -...+ ap-32 - p-2a +…

A: Let a be any solution of x^2 + 1 ≡ 0 (mod p), so that a^2 ≡ −1 (mod p). Becausep doesn't divide a.…

Q: Show that the Bernoulli number B₁ satisfies B₁ = (-1)" ( 2 + 2d²+1) Σ (mod 1) for any positive…

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Q: Let p be an odd prime and r a primitive root of p. Show that if a and b are positive integers in Zp,…

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Q: 3. (i). Suppose f (x) is a polynomial with integer coefficients such that f(2) = 0(mod 5) f(4) =…

A: Given that  (a) f(2)≡0(mod 5)f(4)≡0(mod 9) We have to find a polynomial such that f(x)≡0(mod 45)…

Q: (2) Let p be an odd prime and suppose that m e Z is not divisible by p. (1) Show that E() \p- = 0.…

A: (1) Given that p is an odd prime. The aim is to show that ∑a=1p-1ap=0. Now take, ∑a=1p-1ap,…

Q: (b) Prove by Induction that (3i – 1)(3i +2) = 3n +6n2 +n for n21 i=1

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Q: Fix n e N. Prove that gcd(5n + 3, 3n + 1) = 1

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Q: The kth harmonic number is defined to be Hk = 1 + 1 2 + 1 3 + · · · + 1 k . Prove that for all n ∈ N…

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Q: 9. Let n E Z. Prove that 3k EZ such that n' = 8k or n' = 8k + 1. In other words, prove that by…

A: We will use the fact that any integer is either even or odd to prove the required result.

Q: Let n = pq where p and q are two distinct primes and consider Zn. Further, let [a] E Zn be an…

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Q: QUESTION 3 Prove that 2n > n2 if n is an integer greater than 4. For the toolhar press AIT. E10 /RCL

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Q: . Let b, k and m be positive integers that satisfy gcd(b, m) = 1 and gcd(k, ø(m)) = 1, %3D where…

A: Here GCD(b,m)=1 means they are coprime.

Q: If |G| = pq, where p and q are primes that are not necessarily distinct,prove that |Z(G)| = 1 or pq.

A: Given: G=p.q where p &amp; q are primes To prove: ZG=1 or pq

Q: 3. Euler made the following conjecture: Let a be any natural mumber and p, q primes such that p= ±q…

A: Let a be any natural number....and p,q be odd primes...

Q: Prove that if n = p1a1...prar, where the pi are prime, then σ(n)ϕ(n) = n2(1-p1-a,-1)...(1-pr-a,-1

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Q: Theorem 11.3. If p and q = 2p+1 are primes, then either q | Mp or q |Mp +2, but presented next. not…

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Q: In parts (a)- (e), involve the theorems of Fermat, Euler, Wilson, and the Euler Phi-function. (a)…

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Q: (7) In the ring (Z,+,.), we get n{P:P non trival prime ideal in Z} = ...... (a) o (b) (Z, +, .) (c)…

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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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Q: Problem 6. Find all positive primes p such that n = 1001p satisfies that a" = a mod n for all a.

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