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Number Theory

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

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.

Number theory Question

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

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.

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)

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.

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.

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)

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

Asymptotic Analysis and Fibonacci Number

Q.1 For a number x, with 1< x < p, the number x^n mod p can be computed with at most 2log2 n modulo p multiplications. Asymptotic notation questions Q.2 2^(2n) = O(2^n) Q.3 log*n = O(log*(log n)) Q.4 The sqrt n th Fibonacci number can be computed and written in O(log n) time Please see the attac

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

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 &#955;1...&#955;n 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

Prime Ideals and Irreducible Polynomials

Proofs 1. Let F be a field and p(x) and irreducible polynomial over F. Prove that (p(x)) is a prime ideal of F[x]. 2. If R has no divisors of zero, then neither does R[x].

Solving Polynomials

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.