### Theory of Numbers : Fibonacci Numbers

Prove that F1F2 + F2F3 + F3F4 + ...+ F2n F2n+1 = (F2n+1)^2 - 1.

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Prove that F1F2 + F2F3 + F3F4 + ...+ F2n F2n+1 = (F2n+1)^2 - 1.

Principle of Mathematical Induction, Fibonacci Number Prove that F1F2 + F2F3 + F3F4 + ...+ F2n - 1F2n = (F2n)^2.

Prove that (Fn+1)^2 - Fn Fn+2 = (- 1)^n

Please see the attached file for the fully formatted problems.

Fill in the function table. when f(x) = x^2 , f(x) = 2x - 1 and f(x) = x^2 - 2x + 1 x f(x) -3 -2 -1 2 4.

Express 66 as a product of prime numbers.

Theory of Numbers (IX) Principle of Mathematical Induction Fibonacci Number Prove that

Theory of Numbers (VIII) Principle of Mathematical Induction Fibonacci Number Pro

Suppose that F1 = 1, F2 = 1, F3 =1, F4 = 3, F5 = 5, and in general Fn = Fn-1 + Fn-2 for n ≥ 3 ( Fn is called the nth Fibonacci number.) Prove that F1 + F2 + F3 +...+ Fn = F(n + 2) - 1

Meaning is given to the sum and product of all the natural numbers and then it is shown that: 1+2+3+4+... = -1/12 1*2*3*4*... = sqrt(2*pi). See the attached file.

Prove that 1^3 + 2^3 + 3^3 + ... + n^3 = (1 + 2 + 3 + ... + n)^2

Assume that n is odd and a is a primitive root mod n. Let b be an integer with b ≡ a(mod n) and gcd (b, 2n) =1. Show that b is a primitive root mod 2n.

Use Quadratic Congruences to evaluate the following fractions. See attached file for full problem description.

Quadratic Congruences. See attached file for full problem description. Let p be an odd prime. Complete the proof of the question "For which odd primes p is LS(2, p) = 1?" by showing that if and that if .

Prove that there are no integers x, y, and z such that x^2 + y^2 + z^2 = 999.

See attached file. This problem is from the Text Book Mathematical Methods in the Physical sciences 2nd Edition by Mary. L. Boas. Chapter 12. Section 5 problem number 6

List all irreducible polynomial of degree 2, 3 and 4 over F22juu2332. Prove your assertion. Please see the attached file for the fully formatted problems.

List each of the 15 elements x, x^2,...,x^15 of F2[x]/<x^4 +x + 1> as polynomials of degree at most three over F22.j.uu2332 Please see the attached file for the fully formatted problems.

Let P(z) and Q(z) be complex polynomials of degree m and n respectively, such that Q(z) has distinct roots z1...zn. Assume that n >= m+2 . Prove that P(z1)/Q'(z1) +...+ P(zn)/Q'(zn) (hint: evaluate ∫c P(z)/Q(z) dz for a suitable curve C).

1) How do I show x+1 is primitive? 2) How do I prove x^4+x^3+1 is an irreducible polynomial of degree 4 over Z mod?

17. The first three Legendre polynomials are P0(x) =1 , P1(x) = x and P2(x)=1/2(3x^2-1). If x = cos θ, then P0(cos θ)=1 and P1(cos θ)= cos θ. Show that p2(cos θ)= 1/4(3 cos 2θ +1). Book:- Differential Equations, by Dennis G Zill, page ,number 17.

1. The figure shows the graphs of f(x) = X 3 and g(x) =AX 3.What can you conclude about the value of a? 2. If f(x)= x(x+3)(x-1), use interval notation to give all values of x where f(x)>0. 3. If f(x) =x(x-1)(x-4)2 , use interval notation to give all values of x where f(x)>0. 4. Find the quotient and remainder of f(x) =

I had to prove 4 theorems two of them dealt with Abelian elements, automorphism of R2 under compontentwise addition. I want to keep my original work in tack as possble BUT I would like the follwing corrections made based on the following comments. THESE ARE THE ISSUES THAT NEED TO BE ADDRESSED The paper needs to provide

Bernoulli Polynomials are defined and then various properties are demonstrated. Key words: 1) Contour Integral Definition 2) Euler Numbers 3) Fourier Series 4) Evaluation of Riemann Zeta function for even integers 5) Stirling Numbers of the Second Kind

Chapter 3.1 Derivatives of Polynomials and Exponential Functions. I don't understand how to get the answers provided. Please explain this step by step. Differentiate each function. Y = X2 + 4X +3 V(t)=t2 - 1 Z= A + BeY √ (X) 4√(t3) Y10 Answer Y'=3 √ (x

A geologist you spoke with is concerned about the rate of land erosion around the base of a dam. Another geologist is studying the magma activity within the earth in an area of New Zealand known for its volcanic activity. One of the shortcuts they apply when doing calculations in the field is to use synthetic division. After t

(See attached file for full problem description) The problem is solved by integration and evaluation between the limits and algebraic manipulation

There are problems to test the commutativity, distributive property, definition of natural numbers. Please find the attachment for the problems.

Discuss an element of drinking water where you see a public health gap or need that should be addressed. You may bring up a specific case relevant to your community if you have one, or bring up a global issue.

Show that there are infinitely many algebraic numbers such that . If and are distinct algebraic numbers with = , then what are the possible values of ? Illustrate each value in your list by giving a specific choice of and . Please see the attached file for the fully formatted problems.