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Number Theory

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


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

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.

Real Analysis : The Fundamental Theorem of Arithmetic and Prime Factors

1. Prove the following lemma. Lemma Suppose that m and n are natural numbers > 1 and that p is a prime number. The following statements are equivalent: a. p is a prime factor of m or p is a prime factor of n. b. p is a prime factor of m*n Also Use Theorem: The Fundamental Theorem of Arithmetic. 2. Prove the follow

Number Theroy

Is there a perfect square n^2 such that n^2 = -1 (mod p) for p=3 p=5 p=7 p=11 p=13 p=17 p=19? Can you characterize the primes for which n^2 = -1 (mod p) has a solution?

Legendre Theorem and Prime Factors

Let X and Y be independant random variables that are both equally likely to be either 1,2... (10)^N, where... a) Give a heuristic argument that Qk = 1/k^2Q1. (See attachment for full questions)

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

Number theory

Write the numbers 25, 32, 56 to the base 5. write 47, 68, 127 to the base 2.

Collected Prime Factorization and Multiplicative Functions

B6: a) State a formula for tau (n), the number of divisions of n, in terms of the collected prime factorization of n. b) define the term multiplicative function. c) Suppose that f and g are multiplicative functions. Prove that the function h defined by h(n) = SUM (d|n) f(d)*g(n/d) is also multiplicative. d) Fi

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:

Polynomials and Synthetic Division

1. Identify the polynomial written as a product of linear factors. F(x) = x fourth +10x cubic +35x squared +50x +24 2. Solve z cubic-6z squared + 13z-10 given that 2 + i is a root. 3. Find a polynomial with interger coefficients that has the given zeros. 5, 4i, -4i, i, -i 4. Find all vertical asymptotes of th

Riemann Sums, Taylor Polynomials, Taylor Residuals and Radius of Convergence

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


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

Euler Totient Function (Six Problems)

For this problem it helps to know that: 3x7x13 = 273 (a) Define the Euler Totient function, (SYMBOL) For (b) to (f) please see attached. (PLEASE SEE ATTACHMENT FOR COMPLETE PROBLEM AND PROPER SYMBOLS)

Euclid's Algorithm for Greatest Common Divisor

1. (a) Use the Euclidcan Algorithm to find the greatest common divisor of 13 and 21 (b) Is 13 invertible in Z21? If so, find the reciprocal. (c) Suppose x and yare integers, what is the minimum positive value for 13x+21y? Determine all posible values of (x,y) for which the minimum is obtained. (PLEASE SEE ATTACHMENT FOR

Factor Positive Integrers into Primes

Factor into primes the following positive integers: (a) 25 (b) 4200 (c) 10(to the exponent)10 (d) 19 (e) 1 *Please see attachment for proper citation and complete instructions

A Discussion On Prime Factorization

Find two numbers that have a product of 81 and also have a sum of 30 (use prime factorization for the product) Please see attachment for the formatted question.