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Number Theory

Number Theory is the examination of integers. It is also sometimes called “Higher Arithmetic” because it is a discipline of pure mathematics which studies properties and rules of whole numbers. Some of the most prominent constituents of Number Theory are Prime Numbers and Prime Factorization. Although Number Theory addresses many disciplines in Mathematics, in Algebra, it specifically refers to the study of the algebraic properties of mathematical objects of interest. For example, in the following equation:

X^2+3x+2 = 0

The unknown x is referred to as an algebraic number, as there may not be one specific value associated with it. Thus, it is can be viewed as a generalization for a specific set of rational numbers within the context of the equation. The equation above can be solved using the quadratic equation or more simply, by factorization:

X^2+3x+2 = 0

(x+2)(x+1) = 0

X = -1, -2

Thus, through factorization, it can be seen that x is not just one value, but rather it can be two: x = -1 and x = -2.  Thus, Number Theory in Algebra is concerned with evaluating the properties of x, and by extension, unknown variables that can generalize rational numbers.

When do I need absolute values when simplifying nth roots?

This question arises typically when dealing with the principal square root which is the full name of the function involving the radical. An absolute value can arise from a simplification whenever the index is an even integer. Here are two cases, one when the absolute value is simplified out and one when it is required in the fin

Power series expansion of the given product of exponential factors

By considering appropriate series expansions, prove that the power series expansion of the product of the (infinitely many) exponential factors e^{(x^i)/i}, i = 1, 2, 3, ..., is 1 + x + x^2 ... for |x| <1. By expanding each individual exponential factor in the product and multiplying out, also show that the coefficient of x^1

Logarithmic integral: two forms

Define the logarithmic integral li(x) as the integral of the function 1/(log t) from t = 2 to t = x, where x > 2 and "log" denotes the natural logarithm. (a) Determine constants A and B such that li(x) can be expressed in the following two forms: (i) li(x) = x/(log x) + A + g(x), where g(x) is the integral of the function

Number Theory

Exercise 3 . (3 marks) Decode the following message: "79311601" knowing that the public key is n = 8191 x 65537 = 536813567 and α = 7582663 (I used the correspondence A<-> 01, B <-> 02, ... , Z <-> 26, 0 <-> 30, 9 <-> 39 and worked in base 41 to encode this message.)

Earthquake: Richter and Mercalli Scales

Summarize the source of the calculations for the Richter scale. - Compare the Richter scale and Mercalli scale. In your opinion, explain which is more meaningful in the use of reporting earthquake destruction. - Describe how math and science has made reporting earthquakes more accurate.

Expressions -- Properties of Addition and Multiplication

Match the property to the equation a. Identity Property of Addition b. Identity Property of Multiplication c. Additive Inverse Poperty d. Multiplicative Inverse Property e. Commutative Property of Addition f. Commutative Property of Multiplication g. Associative Property of Addition h. Associative Property of Multiplic

Translating Expressions and Equations

See the attached file. 1. The sum of twice a number and three 2. Twice the sum of a number and three 3. The difference of 37 and 4 times a number 4.The total of five times a number and ten is three 5. Twenty-one s the quotient of three times a number and seven 6. There were roses in the vase. Joan cut some mo

Quadratic Equation Calculations

Project # 1 Work only equation: (a) x^ - 2x - 13 = 0 and (c) x^ + 12x - 64 + 0, but complete all 6 steps (a-f) as shown in the example. Project#2, Please select at least five numbers: 0(zero), two even numbers and two odd numbers. Make sure you organize into separate projects. The assignment must include all the

Algebra: functions and linear system

Part 1 1. Define the word "function." 2. Give an example of a function using a set of at least 4 ordered pairs. The domain will be any four integers between 0 and +10. The range will be any four integers between -12 and 5. 3. Explain why the example models a function. 4. Give an example of at least four ordered pairs tha

Perfect and amicable numbers

1) See attached 2) Show that 28 is a perfect number. 3) What is the amicable number to 17296? 4) Express all 5/11 as the sum of distinct unit fractions. 5) Using modular arithmetic, derive a formula to find all numbers that meet the following conditions: - Divide by 3 the remainder is 1 - Divide by 5 the remaind

Modular Arithmetic - Thinking Mathematically

1) The definition of length, weight, and modulus of a check digit scheme. 2) Choose one of the schemes discussed in the worksheets and describe it in terms of length weight and modulus. Then illustrate the check digit computation on an ID number valid for that scheme (for example on a book or some product you have handy).

Well-Ordering and Division Theorem

32. Show that there is no rational number b/a whose square is 2, as follows: if b^2 = 2a^2, then b is even, so b = 2c, so, substituting and cancelling 2, 2c^2 = a^2. Use that argument and well-ordering to show that there can be no natural number a > 0 with b^2 = 2a^2 for some natural number b. 33. Let m be the least common mu

Management Science - Algorithms

Please explain all reasoning and show all steps to help me better my own understanding of this material. Thanks! See the attachment. Problem A Suppose that you need to take a trip from city 1 to city 9 in below figure in the least amount of time. The figure shows the estimated times it will take you to go from one city to

Computation of Wavefunctions of the Harmonic Oscillator

** Please see the attached file for the full problem description ** Use Gaussian multiplication on the Hermite polynomials in the attached document. These give the un-normalized wave functions for the levels of the harmonic oscillator. The 3rd and 4th Hermite polynomials are, respectively: H_2(x) = 4x^2 - 2 H_3(x) = 8x^

Hash Division Function Numbers

A hash file uses the division function to calculate the bucket number from the given key value. The file was set with 11 buckets numbered 0 to 10, where each bucket can contain at most 3 records. Analyse what happens when records with the following keys are inserted into the hash file. Explain what went wrong and suggest a solut

Prime Numbers in Cryptography

1. Prime numbers are often used in cryptography. Why do you think prime numbers would be more useful for creating codes than composite numbers? 2. Explain a real-world problem that you used math to solve. What mathematical expressions did you use in your problem-solving? Define your variables and explain your expression. 3

C++ Code for Testing Random Number Generator

I am using "Numerical Recipes (3rd Edition)" to create a numerical method that computes random numbers. My problem is that I'd like to use tests such as Chi-Square to test the uniformity/performance of the generator, but I don't know how to do this. You can see my basic number generator in the readable file randomnumbers.

Multiplication of polynomials and evaluating expressions

1. How would you teach the multiplication of polynomials? 2. What four steps should be used in evaluating expressions? Could these steps be skipped or rearranged? Explain your answersr. 3. Do you always use the property of distribution when multiplying monomials and polynomials? Explain why or why not. Also what type

Distinct Natural Number Function

Let s be the function that associates with each natural number the sum of its distinct natural number factors. For example, s(6) = 1 + 2 + 3 + 6 = 12. 1. Calculate s(k) for each natural number k from 1 through 15. 2. Are the numbers SQRT(5), PI, and -6 in the domain of the function s? What is the domain of the fun

division, simplify applications

See attached worksheet 1 -20. 12. Solve for u: -8 = 4/u 13. Solver the following equation for x: 1/x + 3 = -3 14. Solve for y: - 2/y+1 + -6 + 5/y+2 The negative sign is in from of the entire first expression. The negative sign -6 is for the number 6 (if there is more than one solution, separate them with

Dividing Polynomials

1) Two polynomials P and D are given. Use either synthetic or long division to divide P(x) by D(x), and express P in the form P(x) = D(x) · Q(x) + R(x). P(x) = 3x^2 + 4x â?' 1, D(x) = x + 5 P(x) =(x+5)(_____)+______ 2) Two polynomials P and D are given. Use either synthetic or long division to divide P(x) by D(x), and

Number theory and the real number system.

Why does Ellie see these pulses as a sign of intelligent life or what is the significant about the number of pulses? Contact If you could send a long message to such extraterrestrial beings - words, pictures, sounds, music - what would you say? How would you describe us? What would you leave out? Could you communicate intell

1. For each of the following statements, write the contrapositive statement, and prove the original statement by proving its contrapositive: (a) If m^2 + n^2 &#8800; 0, then m &#8800; 0 or n &#8800; 0. 2. What is wrong with the following proof of the conjecture "If n^2 is positive, then n is positive.": Proof: Suppose that n^2 is positive. Because the conditional statement "If n is positive, then n^2 is positive" is true, we can conclude that n is positive.

1. For each of the following statements, write the contrapositive statement, and prove the original statement by proving its contrapositive: (a) If m^2 + n^2 &#8800; 0, then m &#8800; 0 or n &#8800; 0. 2. What is wrong with the following proof of the conjecture "If n^2 is positive, then n is positive.": Proof: Sup

Simple Math Calculations in Real-Life Scenarios

1. Solve the problem. If one book of stamps lasts a family three months. How many books of stamps would be needed to last the family a year? Show your step by step work! Include correct units with your solution. 2. Solve the problem. If one book of stamps lasts a family three months. How many books of stamps would be needed t

Calculation of LCM

The Week One Discussion will concentrate on the mathematical fact that all numbers in our real number system are the product of prime numbers. This fact alone is amazing. a. You will select the ages of two people in your life, one older and one younger. It would be great if the younger person was 15 years old or less.

Solving Polynomials and degree of polynomials.

From the given polynomials, identify the polynomials of degree one. a. 11y2 - 5 - 4y b. (3x2)1/2 + 12 c. 7 - (12)1/2x d. 2x + 13x2 e. 5x + 7y + 8 f. (12)1x1 g. x3 + 2x - 10 h. 3x + 4x - 4 Solve the following: i. 2x = -3x + 9 ii. 3x/5 = -6 iii. y/4 + 2 = 7 iv. 16 = -2x/3 v. Find f(1) for f(x) = 2x3 - 3x2