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# Number Theory

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### 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.

### 23 Polynomials Problems : Solving for Roots, Asymptotes, Word Problems, Finding Equations from Roots, Synthetic Division and Function Composition

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) =

### 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

### College Algebra Problems given in the attachment are to be solved

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

### 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.

### Proving an Equality : Lagrange Interpolating Polynomials

Let x0, x1, ..., xn be distinct points and let (see the attached file for the equation) Show that (see the attached file for the equation) for m = 1,2,...,n and for all x. Please see the attached file for the fully formatted problems.

### Advanced Math : Let p be a positive prime integer

1. Let p be a poistive prime integer. Show that there is no rational number r such that p = r^2. Please see the attached file for the fully formatted problems. ---

### Number Theory and Universal Sets

Need help in proving the following. May need above assertion to prove. (See attached file for full problem description)

### Irreducible Polynomials

Show that there are exactly (p^2-p)/2 monic irreducible polynomials of degree 2 over Z_p, where p is any prime. Using the definition of irreducibility, Theorem: A polynomial of degree 2 or 3 is irreducible over the field F iff it has no roots in F, or Lemma of Theorem: The nonconstant polynomial p(x) an element of F[x] is irr

### Finding Two Prime Factors of a Number

The number n = 15744539 is a product of two prime numbers. Find these two prime numbers if it is also given that φ(n) = 15736560. You may only use elementary functions on your calculator (adding, subtracting, dividing, multiplying, taking the square root). Show your work. See the attached file.

### Number Theory. 400 level. Introductory Course in Undergraduate.

Topics usually include the Euclidean algorithm, primes and unique factorization, congruences, Chinese Remainder Theorem, Hensel's Lemma, Diophantine equations, arithmetic in polynomial rings, primitive roots, quadratic reciprocity and quadratic fields. (See attached file for full problem description)

### Taylor Series and Polynomials: Removable Singularity, Continuity

Consider the function f(x)=x/(ex-1)+x/2. (a) f has a so-called "removeable singularity" at x=0, where it is (so far) undefined. What value should we assign to f(0) to make f continuous at x=0? (b) With this taken care of, f actually has a Taylor series about x=0. Find the first 10 terms or so of this Taylor series (use C

### Complete Residue Systems : Suppose that p is a prime number

Suppose that p is a prime number ≥ 3. We can write 1 + 1/2 + 1/3 + 1/(p-1) = a/b with (a,b) =1. Prove that p divides a. Please see the attached file for the fully formatted problem.

### Complete Residue System : Suppose that p is a prime number

Suppose that p is a prime number. Show that 0, (p-1)!/2, (p-1)!/3, ..., (p-1)!/(p-1) is a complete residue system modulo p.

### One to One Map : Suppose that p is a prime number

Suppose that p is a prime number &#8805; 3 and r1, r2...rp is a complete residue system of modulo p. Prove that r1+r2+...+rp is divisible by p.

### Euclidean algorithm, primes and unique factorization, congruence

Topics usually include the Euclidean algorithm, primes and unique factorization, congruences, Chinese Remainder Theorem, Hensel's Lemma, Diophantine equations, arithmetic in polynomial rings, primitive roots, quadratic reciprocity and quadratic fields. (See attached file for full problem description)

### Polynomials, Integers and Divisibility

Problem 6. Suppose that f(s) = adxd+ad-1xd... is a polynomial with integral coefficients (so a0, a1 . . ,ad E Z). Show that f(n)?f(m) is divisible by n ? m for all distinct integers n and m. See the attached file.

### Infinitely Many Primes Using Euclid's Proof

Prove that there are infinitely many primes of the form 4n ? 1. (Hint: Modify Euclids proof that there are infinitely many primes. First, prove that any number of the form 4n ? 1 has a prime factor which is of the form 4k ? 1). See the attached file.

### Prime Factorization : Proof - Square-Free Integers

Show that every positive integer n can be written in the form n = ab where a is square-free and b is a square. Show that b is thi the largest square dividing n. (A square-free integer is an integer that is not divisible by any square > 1).

### Find Prime Factorization (by hand) of 2006 and 2007.

Find Prime factorization (by hand) of 2006 and 2007.

### Taylor Polynomials : Degrees and Plotting

2. (Taylor polynomials) (a) Write down the Taylor polynomials Pn(x) of degree n = 0, 1, 2,3 for the function f(x) = ln x about the point x = 1. (h) Plot the polynomials Pn(x) and the function f(x) on the interval [0, 3] using Matlab. Describe how the error |f(x) ? Pn(x)| behaves with respect to the point x and the degree n.

### Least common multiple

Find the least common multiple of 2345 and 5236, 10000 and 10001.

### Cryptography

List 4 problems dealt with by cryptography & give real world examples of each. 2 paragraphs please.

### Bezout's Theorem for Polynomials

(See attached file for full problem description) --- Suppose that a(x) and m(x) are relatively prime polynomials in F[x]. Bézout's Theorem guarantees the existence of polynomials u(x) and v(x) in F[x] such that... --- (See attached file for full problem description)