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# Number Theory

### Solve: Simplifying Polynomials

Consider the following expression: Polynomial: (-5t^2 + 2t + 5) + (-8t^2 + 6t) - (9t^2 - 2t -3) Make sure to show all of your work.

### Products of Prime Numbers

Please answer the following: Express 81 as a product of prime numbers.

### Solve: Prime Number Factorization

Showing all of the steps which are involved, express 88 as a product of prime numbers.

### Prime Number Factorization

Prime number factorization Express 45 as a product of prime numbers

### Polynomials - How do I check this by multiplying back?

Please see the attached file for the fully formatted problems.

### Number Theory, Fermat's Theorem

Let p and q be prime number greater than 3. Prove that 24|p^2-q^2

### Algebraic Number Theory

Problem 1: Prove that there are no integers x, y, and z such that x^2 +y^2 + z^2 = 999 Problem 2: Show that square root of 2 cubed is an irrational number. Problem 3: For each of the following pairs a and b, use the division algorithm to find quotient q and remainder r. (a) b=189, a=17

### Polynomials : Roots, Holes and Vertical Asymptotes

#1: f(x)=(1x^3-18x^2+101x-168)/(1x^3-2x^2-41x+42) Find: All Roots:? All Holes:? All Vertical Asymptotes:? The answer to the 1st question is NOT 8 or -8. #2: f(x)= (-2x^3+37x^2-222x+432)/(1x^3+3x^2-64x-192) I need the roots: (I have two of them (9/2) and (6) but I need the third)? I need to know the hol

### Simplifying Polynomials and Collecting Like Terms

Section 4.3 Collect like terms. 4 4 52. 3a - 2a + 2a + a Collect like terms and then arrange in descending order. 3 3 4 66. -1 + 5x - 3 - 7x + x + 5 Classify the polynomial as a monomial, binomial, trinomial, or none o

### Binary Representations and Prime Factors

1- For n belongs to N (set of natural numbers) let B(n) denote the number of digits used in the binary representation of n. For example B(1) = 1; B(2) = 2; B(3) = 2; B(4) = 3: Find a closed formula for B(n) for an arbitrary n belongs to N. 2: Prove that if gcd(a, b) = d then a/d and b/d are relatively prime. 3- Find

### Theory of Numbers : Fibonacci Numbers

Prove that F1F2 + F2F3 + F3F4 + ...+ F2n F2n+1 = (F2n+1)^2 - 1.

### Theory of Numbers : Fibonacci Numbers

Principle of Mathematical Induction, Fibonacci Number Prove that F1F2 + F2F3 + F3F4 + ...+ F2n - 1F2n = (F2n)^2.

### Theory of Numbers : Fibonacci Numbers

Prove that (Fn+1)^2 - Fn Fn+2 = (- 1)^n

### Taylor Polynomials : Finding nth Terms

Please see the attached file for the fully formatted problems.

### Polynomials: Complete the Function Table and Find the Values

Fill in the function table. when f(x) = x^2 , f(x) = 2x - 1 and f(x) = x^2 - 2x + 1 x f(x) -3 -2 -1 2 4.

### Express 66 As Prime Number Products

Express 66 as a product of prime numbers.

### Solution to "Mathematical induction" question

Theory of Numbers (IX) Principle of Mathematical Induction Fibonacci Number Prove that

### Theory of Numbers (VIII): Principle of Mathematical Induction: Fibonacci Number

Theory of Numbers (VIII) Principle of Mathematical Induction Fibonacci Number Pro

### Theory of Numbers : Fibonacci Number

Suppose that F1 = 1, F2 = 1, F3 =1, F4 = 3, F5 = 5, and in general Fn = Fn-1 + Fn-2 for n &#8805; 3 ( Fn is called the nth Fibonacci number.) Prove that F1 + F2 + F3 +...+ Fn = F(n + 2) - 1

### Regularized Sum and Product of the Natural Numbers

Meaning is given to the sum and product of all the natural numbers and then it is shown that: 1+2+3+4+... = -1/12 1*2*3*4*... = sqrt(2*pi). See the attached file.

### Theory of Numbers : Principle of Mathematical Induction - Proof

Prove that 1^3 + 2^3 + 3^3 + ... + n^3 = (1 + 2 + 3 + ... + n)^2

### Theory of Numbers - Primitive Root

Assume that n is odd and a is a primitive root mod n. Let b be an integer with b &#8801; a(mod n) and gcd (b, 2n) =1. Show that b is a primitive root mod 2n.

### Theory of numbers..

Use Quadratic Congruences to evaluate the following fractions. See attached file for full problem description.

### Quadratic Congruences - Theory of Numbers

Quadratic Congruences. See attached file for full problem description. Let p be an odd prime. Complete the proof of the question "For which odd primes p is LS(2, p) = 1?" by showing that if and that if .

### Number Theory : Prove that there are no integers x, y, and z such that x^2 + y^2 + z^2 = 999.

Prove that there are no integers x, y, and z such that x^2 + y^2 + z^2 = 999.

### Use of the generating function for Legendre polynomials

See attached file. This problem is from the Text Book Mathematical Methods in the Physical sciences 2nd Edition by Mary. L. Boas. Chapter 12. Section 5 problem number 6

### Irreducible Polynomials and Fields

List all irreducible polynomial of degree 2, 3 and 4 over F22juu2332. Prove your assertion. Please see the attached file for the fully formatted problems.

### Fields and Polynomials

List each of the 15 elements x, x^2,...,x^15 of F2[x]/<x^4 +x + 1> as polynomials of degree at most three over F22.j.uu2332 Please see the attached file for the fully formatted problems.

### Complex Polynomials, Roots and Residue Theory

Let P(z) and Q(z) be complex polynomials of degree m and n respectively, such that Q(z) has distinct roots z1...zn. Assume that n >= m+2 . Prove that P(z1)/Q'(z1) +...+ P(zn)/Q'(zn) (hint: evaluate &#8747;c P(z)/Q(z) dz for a suitable curve C).