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# Number Theory

Addition of radicals are treated similar to polynomials, but instead of multiples of x's and x2's, we have multiples of things that look very much like root x and root x - 2. When adding such expressions together, no arithmetic can be done underneath the radical. However, like radicals can be combined together. Question 1)

### Bessel functions

Problem: (1) Show that there exist exist polynomials P_n(z) and Q_n(z) for positive integers n such that the Bessel function J_{n + 1/2}(x) satisfies J_{n + 1/2}(x) = P_n(x^{-1/2}) sin x + Q_n(x^{-1/2}) cos x. (2) Find ( P_1(z) ) and ( Q_1(z) ). This problem tells us that Bessel functions of half-integer order can b

### Special values of Chebyshev polynomials

I need to use the generating function for Chebyshev polynomials to show the special values. I am easily able to show these using T_n(x) = cos(ncos^-1(x)) but cannot figure it out using the generating function. I have tried plugging x = 0, or -1, or 1 into both sides of the generating function and using the geometric series sum f

### 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

### Irreducible polynomial problems

Determine the number of monic irreducible polynomials of degree 4 in F_q without using the Moebius Inversion Formula.

### 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.

### Divisibility Properties of Various Products of Integers

Prove that if x + y is even, then the product xy(x + y)(x - y) is divisible by 24, and that without this restriction, 4xy(x- y)(x + y) is divisible by 24. Consider that any integer is of the form 3k, 3k + 1, or 3k + 2 in showing that 3|xy(x + y)(x - y). Similarly, because any integer is of the form 8k, 8k + 1, ..., or 8k + 7,

### Lottery probability in numbers

What is the probability that 6 numbers drawn in a lottery will be prime? The numbers range from 1-46. So i know that the first prime numbers under 46 are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43. Which is a total of 14. Would it be something like 46!/(14!)(32!)? I don't understand probability that well.

### Number Theory: Birthday Problem

Hello, I haven't seen this before and I'm having trouble figuring it out. There are 10,000 people in a stadium. Show that there is at least of group of 28 people with the same birthday. Thanks for any help -john

### Divided Difference Interpolation

2. Use Eq. (3.10) to construct interpolating polynomials of degrees one, two, and three for the following data. Approximate the specified value using each of the polynomials. a. f(0.43) if f(0)=1,f(.25)=1.64872 f(.5)=2.71828,f(.75)=4.48169 b. f(0) if f(-.5)=1.93750,f(-.25)=1.33203,f(.25)=.800781,f(.5)=.687500

### Presentation of Sequences on TI-9 and Related Calculators

This is a 17:38 annotated video of doing basic arithmetic and geometric sequence manipulations on the TI-89 calculator: Finding part of a sequence, one item in a sequence, or the sum of any part of the sequence. Explanatory remarks touch on common difference and common ratio. Some remarks and images are made to make the video us

### Prime Factorization Using Factor Fireworks

Please provide step-by-step instruction to help me solve the following problems. I 1. Make two different Factor Fireworks for 64 and 84. 2, Write an equation that shows 64 as a product of prime numbers. Then use the equation to help find 54/16. 3. Write an equation that shows 84 as a product of prime factors. Th

### flaw with Euclid's first proposition

Where is the flaw and how can it be fixed? If R^2 is taken as a model for the Euclidean plane, does this help resolve the difficulty? Explain.

### Set of all finite and cofinite subsets of N

A subset A of the set N of all natural numbers is defined to be cofinite if N - A (the complement of A in N) is finite. Show that the set of all finite and cofinite subsets of N is an algebra but not a sigma-algebra.

### Discussing irreducible polynomials

a. Let F be a field and f (x) an irreducible polynomial of degree 3 in F [x]. Show that if K is an extension of F of dimension 10, then f(x) is irreducible in K[x]. b. Let F be a field and f(x) an irreducible polynomial of degree 5 in F[x]. Show that if K is an extension of F of dimension 7, then f(x) is irreducible in K[x].

### 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

### Applying Euler's Totient Theory to Cryptography

Please show/discuss extensively how number theory is applied in cryptography.

### prime factorization of an integer

In the prime factorization of an integer, what is the maximum number of prime factors greater than the square root of that integer? If there is a prime factor greater than the square root of that integer, can the integer be a perfect square?

### There are 130 cabinets in a row in a warehouse. The first worker to arrive opens the door of every cabinet.

How is this question solved? There are 130 cabinets in a row in a warehouse. The first worker to arrive opens the door of every cabinet. The next employee visits every other door and closes it. The third arrives and visits every third door and opens it if it's closed, or closes it if opened. The fourth visits every fourth

### Prime and composite numbers

Please respond to the following questions and provide reference if possible. Thank you. a) What is a prime number, and why are prime numbers important? b) What do you call a number that is not prime? c) What is the largest prime number ever found? d) Is the number 294,822,345,711 prime? How about 173,658,965?

### Countable/Uncountable Sets

Please see the attached image for questions I and II. III) If A is a countable subset of an uncountable set X, prove that X A is uncountable. IV) Suppose that f is a function from X into Y so that the range of f is uncountable. Prove that X is uncountable. V) Prove that the set of all polynomials with rational coeffici

### Product of primes

How can you prove that 640,000,000 is a product of primes without actually finding the primes? Explain your process.

### 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

### Number theory: Euler Totient function (phi) and Euler-Fermat Theorem

Q12: (i) Calculate phi(15) in THREE ways. (ii) Express in modular arithmetic [hint:the number of integers from 1 to m that are relatively prime to m is denoted by phi(m). it is the number of elements in the set a:1=a=m and gcd(a,m)=1 ]

### 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

### Short essay on ranked data

A common error is neglecting to get ranked data for some statistical processes. Why do we have to have ranked data? Give an example of where you would need ranked data. (short paragraph answer, 100 words)

### 5 Questions and solutions about data, data rates, data storage and data compression

This is more along the lines of computer driven word questions, but figured my best bet for asisstance would be the Math section. That is why it is being posted here. Answer ONLY 5 of the questions attached for 6 custom credits. Pick any 5

### A perfect number

A perfect number is an integer that equals the sum of its proper positive factors. For example, the proper positive factors of 6 are 1, 2, and 3. Since 1 + 2 + 3 = 6, we refer to 6 as a perfect number. Similarly, 28 = 1 + 2 + 4 + 7 + 14, so it is also a perfect number. I intend to show that the sum of the reciprocals of the