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Number Theory

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

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.

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.

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

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)

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).

Cryptography

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

Prime numbers

6301 is prime. If x, y, and z are integers that are not divisible by 6301, which of the following is equal to x^6299.y^12600.z^18903 mod 6301 ? (a) xyz (b) yz2/x2 (c) z3/x (d) 1/(x2 y2) (e) none of the above

Primality

Which of the following is true. a) 16 is a non-trivial square root of 1 modulo 51; hence 51 is composite b) 7 is a non-trivial square root of 1 modulo 47; hence 47 is composite c) 8 is a non-trivial square root of 1 modulo 55; hence 55 is composite d) all of the above e) None of the above

Taylor Series and Polynomials

Using the fact that 1+x = 4+(x-3), find the Taylor series about 3 for g. Give explicitly the numbers of terms. When g(x)=square root of 1+x Check the first four terms in the Taylor series above and use these to find cubic Taylor polynomials about 3 for g. Use multiplication of Taylor series to find the quartic Taylor polyn

Prime and Nonprime Numbers : Multiplication

Does a prime number multiplied by a prime number ever result in a prime - Why? Does a nonprime multiplied by a nonprime ever result in a prime - why? Is it possible for an extremely large prime to be expressed as a large integer raised to a very large power? Explain. Are there infinitely many natural numbers that are not pri

Taylor Polynomials

The problem is from Numerical Methods. Please show each step of your solution and tell me the theorems, definitions, etc. if you use any. Show that... Taylor polynomials... Please see attached.

Solve and simplify

1. Use synthetic division to determine if the first set of numbers are zeros of the given polynomial a. -3, 2. f(x) = 3x³ + 5x2 - 6x + 18. a. -4, 2. f(x) = 3x3 + 11x2 - 2x + 8. 2. Given the polynomial f(x) = 2x 3 -5x2-4x+3, find the solutions if the function is completed as a) f(x) =0 b) f(x+2)=0 d) f(2x) = 0 3. To

Congruences

5. A polynomial is said to be monic is its leading coefficient is perpendicular... Please see attached.

Proofs : Pairwise Real Numbers, Natural and Irrational Numbers

Problem 1. Let n be a natural number and a1.... ,an > 0 be pairwise different positive real numbers. Show that if λ1...λn are such real numbers that the equality ... holds true for all x E R then .... Problem 2. Show that there are infinitely many real numbers x in the interval [0, pi/2] such that both sinx and co

Finding area & factoring

I am trying to figure a problem that involves the area of a deck. If a deck is rectangular and has an area of X2 + 6x + 8 (the 2 is squared) square feet and a width of x + 2 feet. I am trying to determine the lengths of the deck.

Solving Polynomial Functions

I hope you will be able to help me with this problem, I'm really stuck. 10-(k+5) = 3(k+2) I would also appreciate any explanation.

Explain the Proof Step-by-step : Irrational Function

Recall that a perfect sqaure is a natural number n such that n = (k^2), for some natural number k. Theorem. If the natural number n is not a perfect square, then n^(1/2) is irrational. Proof. S(1): Suppose n^(1/2) = r/s for some natural numbers r and s. S(2): We may assume that r and s have no prime factors in common,

Prove that there is a one-to-one correspondence between the power set of a countably infinite set A and the set S of all countably infinite sequences of 0's and 1's, and that the power set of A is an uncountable set.

For any set B, let P(B) denote the power set of B (the collection of all subsets of B): P(B) = {E: E is a subset of B} Let A be a countably infinite set (an infinite set which is countable), and do the following: (a) Prove that there is a one-to-one correspondence between P(A) and the set S of all countably infinite seq

Real Life Applications of Complex or Imaginary Numbers

When solving a quadratic equation using the quadratic formula, it is possible for the b2 - 4ac term inside the square root (the discriminant) to be negative, thus forcing us to take the square root of a negative number. The solutions to the equation will then be complex numbers (i.e., involve the imaginary unit i). Question:

Asymptotes, Synthetic Division and Even and Odd Power Theorem

1. FIND ALL VERTICAL ASYMPTOTES OF THE FUNCTION. *******X+5 F (x) ------------------- ****4x squared+7 2. WHICH SHOWS THE TRUE STATEMENT FOR THE GRAPH OF THE RATIONAL FUNCTION g. *******X+2 g(x) ------------------- ****x squared+2x-3 3.USE SYNTHETIC DIVISION TO FIND UPPER AND LOWER BOUNDS OF THE REAL ZEROS OF f.

Riemann Sums, Taylor Polynomials, Taylor Residuals

Please see the attached file for the fully formatted problems. 1. ? Calculate the Taylor Polynomial and the Taylor residual for the function . ? Prove that as , for all . ? Find the Taylor series of f. ? What is the radius of convergence for the Taylor series? Justify your answer. 2. ? Let f:[0,1] be a bo

Linear Equations and their Solutions

From the given polynomials, identify the polynomials of degree one. a. 311y - 5 - 43y b. (11y2)1/4 + 14 c. 10 + 19x - 2x2 d. 2 + 15x e. 52y4 + 7x + 2 f. (68)1y1 g. x3 + 3x - 9 Solve the following: i. -2x = 3x + 4 ii. 3x/4 = 6 iii. y/6 + 1 = 9 iv. 6 = -2x/4 v. Find f(1) for f(x) = 4x3 - 3x2 - x + 2

Polynomials

A. Solve the following questions involving fundamental operations on polynomials a.Find p(x) + 4q(x) given p(x)=4x4 + 10x3 - 2x2 + 13 and q(x) = 2x4+ 5x2 - 3 b. Find P(-1/2) if P(x) = 2x4 + x3 + 12 c. Simplify: (-4 + x2 + 2x3) - (-6 - x + 3x3) - (-6y3 + y2) d. Add: (2x2 + 6y2 + 4z2 + 3xy + yz + zx) + (4x2 + 3y2