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Number Theory

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

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

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.

Complex numbers questions

1. The equation X^5 - 2X^4 - X^3 + 6X - 4 = 0 has a repeated root at X=1 and a root at X-2. By a process of division and solving a quadratic equation, find all the roots and hence write down all the factors of X^5 - 2X^4 - X^3 + 6X - 4 2. Given that cosX= (e^jx + e^-jx)/2

Taylor Polynomials

F(x) = ln5 + ln(1-1/5x) Using substitution in one of the standard Taylor series, find the Taylor series about f for 0. Give all terms up to the term in x^3.

Game Theory : Two-Player Card Game

In the two-player game of Two Stacks, a deck of cards (with the joker added, for a total of 53 cards) is randomly divided into two piles. The two players take turns removing cards from one pile or the other. On a player's turn, that player may remove any positive number of cards from a single pile. The object of the game is to r

Proof : Polynomials

Problem: Let f(x) and g(x) be nonzero polynomials in R[x] and assume that the leading coefficient of one of them is a unit. Show that f(x)g(x) doesn't equal 0 and that deg[f(x)g(x)] = deg(f(x)) + deg(g(x))

Number Theory

Find all the integers such that when the final digit is deleted the new integer divides the original one. Can you generalize this to deleting other digits?

Cryptology

Decipher the following CFQGE KAZEMF ZMAGVMC NMO VYSV which was obtained by a formula of the type y=kx (mod 26)