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Number Theory

Theory of Numbers : Fibonacci Number

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

Theory of Numbers - Primitive Root

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.

Theory of numbers..

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

Quadratic Congruences - Theory of Numbers

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 .

Irreducible Polynomials and Fields

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.

Fields and Polynomials

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.

Complex Polynomials, Roots and Residue Theory

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 &#8747;c P(z)/Q(z) dz for a suitable curve C).

Primitive polynomials

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?

Legendre Polynomials

17. The first three Legendre polynomials are P0(x) =1 , P1(x) = x and P2(x)=1/2(3x^2-1). If x = cos &#952;, then P0(cos &#952;)=1 and P1(cos &#952;)= cos &#952;. Show that p2(cos &#952;)= 1/4(3 cos 2&#952; +1). Book:- Differential Equations, by Dennis G Zill, page ,number 17.

Abstract Algebra (4 year College)

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 Properties

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

Derivatives of Polynomials and Exponential Functions

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

Synthetic Division, Functions, and Interest

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

Legendre equation

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

Public health / drinking water safety

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.

Algebraic Numbers

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.