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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

### Prove that if p is a prime number of the form 4n + 1

Prove that if p is a prime number of the form 4n + 1, then the following congruence holds: x^2 is congruent to -1 mod p. The more detail problem is in the a

### Mordell Equations

Exercise 5. The aim of this exercise is to prove that the Mordell equation y^2 = x^3 - 5 has no solutions. We proceed by contradiction and assume that (x, y) is an integral solution. 1. By reducing Mordell equation mod 4, show that y is even and x = 1 mod 4. 2. Show that y^2 + 4 = (x-1)(x^2 + x +1) 3. Show that x^2 + x +

### Numerical Example of Encryption Using the RSA Method

Please see the file below about encoding a birthday. For example the birthday is 250692, how to calculate? Exercise 2: Encode your date of birth (format DDMMYY) using the public key: n = 536813567 a = 7582663 (Use the correspondence 0 <-> 0, 1 <-> 1, ..., 9 <-> 9 and work in base 10 to encode this message.)

### Numerical Solution of Mordell's Equation using Number Theory

See the attached question on proving the Mordell equation has no solutions using integral solutions.

### 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.)

### Prove that if c  is odd ,  then  (ab/c) = (a/c)(b/c)

Prove that if c  is odd ,                                             then

### Euler's Criterion to Determine Quadratic Residue Modulo p

Use Euler's Criterion to determine whether a is a quadratic residue modulo p in each of the following instances: (a)  a = 2,  p = 5;    (b)   a = 4,  p = 7;                                       (c)  a = 3,  p = 11;    (d)   a = 6,  p = 13.

### Set up a linear equation for internet service bill

An internet service provider charges \$25 for a connection fee and then \$16 per month. Write an expression to model the total cost and then evaluate the expression for 1 to 5 months of internet access. See the attached file for the full problem description.

### Expected Value and Tree Diagram

-Provide one example to show how you can use the Expected Value computation to assess the fairness of a situation (probability experiment). Provide the detailed steps and calculations. -Develop a tree diagram for tossing two, eight-sided gaming dice to figure out how many possibilities there are. -Discuss the purpose of using

### 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.

### Exponential and Logarithmic Models

http://earthquake.usgs.gov/. "The earthquake with a magnitude 8 releases a million times more energy than an earthquake with magnitude 4." Assess the accuracy of this statement. Explain your answer in mathematical terms.

### 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

### Generating Pythagorean Triples

Using the below scenarios: 1. Build or generate at least five more Pythagorean Triples using one of the many formulas available online for doing this. 2. After building your triples, verify each of them in the Pythagorean Theorem equation The numbers 3, 4, and 5 are called Pythagorean triples since 32+42=52. The numbers 5,

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

### Normalization of Hermite Polynomials

Knowing that {Hn}, n = 0, 1, 2... are orthogonal and (Ho, Ho) = sqrt pi, (Hn, Hn) = 2n(Hn-1, Hn-1), nEN with respect to a suitable weight function p(x), write down first two orthonormal Hermite polynomials. Please see attachment for proper formatting.

### Non-Negative Residue-Modulo

Could you please help explain these problems?: 16. Find the least non-negative residue of: (i) 5^18 mod 11; (ii) 4^47mod 12; 28. Show that 11 divides 10a+b if and only if 11 divide a - b. Use this to show that 11 divides 232595. 30. Find the lease non-negative residues mod 7, 11 and 13 of 58473625.

### Unique Factorization Prime/Co-Prime

1. Prove that a and b are coprime if and only if no prime numbers divides both a and b 2. Show that [a,m] = m if and only if a divides m

### 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

### Factoring into Prime Factors

The number 160451 is obtained by multiplying two prime numbers. Find these prime numbers.

### The Only Prime Triplets

A classic unsolved problem in number theory asks if there are infinitely many pairs of `twin primes', pairs of primes separated by 2, such as 3 and 5, 11 and 13, or 71 and 73. Prove that the only prime triple (i.e. three primes, each 2 from the next) is 3, 5, 7.

### 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

### Induction on a Sum of Natural Numbers

Let f:N x N -> N be the function defined recursively as follows: f(0, 0) = 6 f(i, j) = f(i - 1, j) + 2 if i > 0 and j = 0 f(i, j) = f(i, j - 1) + 1 if j > 0 Use induction on the sum i + j to prove that f(i, j) = 2i + j + 6 for all (i, j) in N x N.

### Meaning of Notation

I am struggling with the notation/language of math. 1. What does f: Z----> Z or f: A--->A or f: N--->N mean? Are these the set of integers, natural numbers or any set in general (it is not the complex set)? 2. If I have: Let A = {n:n>2} f: A--->A followed by some function does this mean A is a set of n integers?