Let p be a prime divisor of the Fermat number Fn 22" + 1. : 2n+1 (a) Show that ord, 2 (b) Use part(a) to show that 2"+1 | (p – 1), and conclude that p must be of the form 2"+1k +1.

Q: (b) Prove by induction that (3i – 2)² = n(6n² – 3n – 1) for n 21 %3D i=1

A: Prove the equation ∑i=1n3i-22=12n6n2-3n-1 for all n≥1. Prove the equation for n=1. Substitute n=1…

Q: k) Prove that x = 5108 = 179 (mod 433) is a solution to the equation x² + 1 = 0 (mod 433), and use…

A: 52 ≡25(mod 433)54 ≡625≡192(mod 433)55 ≡960≡94(mod 433)56 ≡470≡37(mod 433)512 ≡1369≡70(mod 433)524…

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: 4. Prove by mathematical induction that for every positive integer n, II (G- 1 %3D 3i-1 31 3i+1…

A: We have to prove given expression using mathematical induction.

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Q: For any n > 1, prove that the irreducible factorization over Z ofxn-1 + xn-2 + . . . +x +1 is π…

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Q: Prove by mathematical induction that j=1 for all positive integersn. AI

A: To prove:                                                                        ∑j=1nj2≥∑j=1nj2,…

Q: & Give an example of a polynomial (x) € Z[] and a prime p such that f(x) is reducible in Qx] but…

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Q: 5. Prove by induction that for positive integers n, E-1(3i² + 2i + 5) = ÷(2n² + 5n² + 13n).

A: Given statement is : ∑i=1n(3i2+2i+5)=12(2n3+5n2+13n) Step1: Let us check if or not the given…

Q: Now let ap be a nonzero element in Z/pZ. Show that a, is invertible. (Hint: Invoking the fact that a…

A: Let [a]p be a nonzero element in Z/pZ.

Q: Let p be a prime divisor of the Fermat number Fn= 22^n+ 1 i. Show that ordp2 = 2n+ 1 ii. Use…

A: Given p be a prime divisor of the number Fn=22n+1. (i) Since p divides Fn=22n+1, therefore: Fn ≅0…

Q: x4 >is ........in Q[x] Not prime ideal ONone of the choices O Prime not maximal

A: Answer is maximal ideal , i.e , last option , and proof is given below.

Q: (b) Prove that -1+²+...+x+ 1 is irreducible over Z if and only if n is a prime

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Q: Prove that k() = n(1) (n = 1) for integers n and k with 1 ≤ k ≤ n, using a: a) combinatorial proof;

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Q: 2. Let a be a positive integer. Explain why the linear congruence az b (mod a + 1) has a solution in…

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Q: Exercise 5.11. Consider the natural numbers 2 and 7, which happen to be relatively prime. Find…

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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: with Zeros of 2i with integer coeRficients that haf E Puno

A: We need to find the polynomial of degree three with integer coefficient

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

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

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Q: 2) Prove that vnEN 12n+1 is in lowest term, i.e., prove that 30n+1 gcd(12n + 1,30n + 1) = 1 3) Vns A

A: As per our guidelines, we are supposed to solve only first question. Kindly repost other question as…

Q: Find the order of [5] in Z23 using Fermat's Theorem.

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Q: 7. Prove by induction. For any n2 0, E F = F,Fn+1• k=0

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Q: Prove that F2 + F = F2n+2 where Fn is the nth Fibonacci number. %3D n+1

A: Given, Fn2+Fn+12=F2n+2 As we know that, F1=1,F2=1,F3=2,F4=2,F5=4,F6=5

Q: 6. Prove that 2n +5 and 3n+7 are relatively prime for every integer n.

A: What is Relatively Prime Numbers: Relatively prime or coprime numbers are two such numbers which…

Q: Prove that Pn(1) = 1 and Pn (-1) = (-1)"

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Q: (b) Prove that there are infinitely many prime numbers of the form 6n + 5

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Q: Number theory

A: Consider the provided question,

Q: Factor 19 into irreducible elements in the Gaussian integers, Z[i].

A: We have to factor 19 into irreducible elements in the Gaussian integers, ℤ[i]. Irreducible element :…

Q: 1 (b) Prove by induction that (3i – 2)² = ;n(6n² – 3n – 1) for n 2 1

A: We have to prove ∑i=1n3i-22=12n6n2-3n-1 for n≥1 by using mathematical induction.

Q: Let n E Z. Prove that n3 – 7n + 18 is divisible by 6 for every interger n. Do not use induction.

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Q: 2. List all the polynomials in Z3[x] that have degree 2.

A: To find all the polynomials in Z3[x] that have degree 2

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Q: 4. Prove that f+ f+..+f = fnfn+1, where n is a positive integer and fn is the nth Fibonacci number.

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

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Q: 2. (a) Define the Euler phi-function and use its properties from the lectures to prove that for all…

A: Euler's phi function is one of the most important number theoretic functions. For any positive…

Q: 10 a. Show that 2 is equal to the product of a unit and the square of an irreducible in Z[i). b.…

A: We will give the result in next step. Note that the ring given is called ring of Gaussian integers.

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Q: infinitely many primes of the form

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Q: Exercise 1. Show that there are infinitely many primes of the form 4n +3. Hint: you might find it…

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Q: Let x, y and n denote integers. Use induction to prove that Vn 2 5: 3x, y > 0 such that n= 2x + 3y…

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Q: Prove that there are infinitely many primes of the form p = 4n + 3.

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Q: Find all monic irreducible polynomials of degree 2 over Z3.

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Q: Let q1, q2, q3, . . . be defined by q1 = 2, qn = 3qn−1 − 1 for n ≥ 2. Show by induction that for all…

A: We are given a sequence of qn and we have to use induction to prove that : qn = 1/2(3n + 1)

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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Q: 1. Find the least no EN such that 10k + 3 no, it has the stated property. and prove that

A: Given: 10k+3<2k, ∀k≥n0, where n0∈ℕ is a least one. To find: Value of n0. To prove: 10k+3<2k,…

Q: Take non-zero positive integers m and n. Show that Z/mZx Z/nZZ/mnZ if and only if m and n are…

A: Given That:To take non-zero integers m and n .   ℤmℤ × ℤnℤ ≅ ℤmnℤ if m and n are relatively prime ,…