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# Number Theory

### Prime numbers

6301 is prime. If x, y, and z are integers that are not divisible by 6301, which of the following is equal to x^6299.y^12600.z^18903 mod 6301 ? (a) xyz (b) yz2/x2 (c) z3/x (d) 1/(x2 y2) (e) none of the above

### Primality

Which of the following is true. a) 16 is a non-trivial square root of 1 modulo 51; hence 51 is composite b) 7 is a non-trivial square root of 1 modulo 47; hence 47 is composite c) 8 is a non-trivial square root of 1 modulo 55; hence 55 is composite d) all of the above e) None of the above

### Taylor Series and Polynomials

Using the fact that 1+x = 4+(x-3), find the Taylor series about 3 for g. Give explicitly the numbers of terms. When g(x)=square root of 1+x Check the first four terms in the Taylor series above and use these to find cubic Taylor polynomials about 3 for g. Use multiplication of Taylor series to find the quartic Taylor polyn

### Asymptotic Analysis and Fibonacci Number

Q.1 For a number x, with 1< x < p, the number x^n mod p can be computed with at most 2log2 n modulo p multiplications. Asymptotic notation questions Q.2 2^(2n) = O(2^n) Q.3 log*n = O(log*(log n)) Q.4 The sqrt n th Fibonacci number can be computed and written in O(log n) time Please see the attac

### 20 Fractions Notations Problems: Multiplication and Division (Please Check My Answers)

1. Find the prime factorization of 54. ____a. 2.2.2.3 _x__b. 2.3.3.3 ____c. 2.2.17 ____d. 2.2.3.3 2. Determine which of the following is divisible by 3. ____a. 3106 ____b. 2251 _x__c. 1239 ____d. 1172 3. Determine which of the following is divisible by 8. __x_a. 1336 ____b. 1473 ____c. 1534 ____d.

### Prime and Nonprime Numbers : Multiplication

Does a prime number multiplied by a prime number ever result in a prime - Why? Does a nonprime multiplied by a nonprime ever result in a prime - why? Is it possible for an extremely large prime to be expressed as a large integer raised to a very large power? Explain. Are there infinitely many natural numbers that are not pri

### Taylor Polynomials

The problem is from Numerical Methods. Please show each step of your solution and tell me the theorems, definitions, etc. if you use any. Consider the function f(x)=log(1+x) a. Write down the Taylor polynomial of degree n... b. How large must n be in order to approximate log(1.1)... Please see attached.

### Taylor Polynomials

The problem is from Numerical Methods. Please show each step of your solution and tell me the theorems, definitions, etc. if you use any. Show that... Taylor polynomials... Please see attached.

### Number Theory: RSA Enciphering Exponent

1) Let e = ... be an RSA enciphering exponent. Prove that, for any... Please see attached.

### Solve and simplify

1. Use synthetic division to determine if the first set of numbers are zeros of the given polynomial a. -3, 2. f(x) = 3x³ + 5x2 - 6x + 18. a. -4, 2. f(x) = 3x3 + 11x2 - 2x + 8. 2. Given the polynomial f(x) = 2x 3 -5x2-4x+3, find the solutions if the function is completed as a) f(x) =0 b) f(x+2)=0 d) f(2x) = 0 3. To

### The Mobius Inversion Formula 1) Prove that &#8721; &#956;(d) &#966;(d) = &#928; (2 - p) d/n p/n Primitive roots modulo p 2) Find all primitive roots modulo 5, modulo 9, modulo 11, modulo 13 and modulo 15. Prime Numbers 3) The Fermat numbers are numbers of the form 2^(2^n) + 1 = &#966;n . Prove that if n < m, then &#966;n| &#966;m - 2 4) Prove that if n &#8800; m, then gcd (&#966;n, &#966;m) = 1 5) Use the above exercise to give a proof that there exist infinitely many primes.

Theory of Numbers Mobius Theorem

### Congruences

5. A polynomial is said to be monic is its leading coefficient is perpendicular... Please see attached.

### Proofs : Pairwise Real Numbers, Natural and Irrational Numbers

Problem 1. Let n be a natural number and a1.... ,an > 0 be pairwise different positive real numbers. Show that if &#955;1...&#955;n are such real numbers that the equality ... holds true for all x E R then .... Problem 2. Show that there are infinitely many real numbers x in the interval [0, pi/2] such that both sinx and co

### Prime Ideals and Irreducible Polynomials

Proofs 1. Let F be a field and p(x) and irreducible polynomial over F. Prove that (p(x)) is a prime ideal of F[x]. 2. If R has no divisors of zero, then neither does R[x].

### Measure Theory : Continuity and the Monotone Convergence Theorem

Let f be a nonnegative integrable function. Show that the function F defined by ... is continuous by using the Monotone Convergence Theorem. Please see the attached file for the fully formatted problems.

### Finding area & factoring

I am trying to figure a problem that involves the area of a deck. If a deck is rectangular and has an area of X2 + 6x + 8 (the 2 is squared) square feet and a width of x + 2 feet. I am trying to determine the lengths of the deck.

### Solving Polynomial Functions

I hope you will be able to help me with this problem, I'm really stuck. 10-(k+5) = 3(k+2) I would also appreciate any explanation.

### Powers of Prime Factors

Write 4^-2 x &#8730;16 x 3/12^-1 as powers of prime factors.

### Explain the Proof Step-by-step : Irrational Function

Recall that a perfect sqaure is a natural number n such that n = (k^2), for some natural number k. Theorem. If the natural number n is not a perfect square, then n^(1/2) is irrational. Proof. S(1): Suppose n^(1/2) = r/s for some natural numbers r and s. S(2): We may assume that r and s have no prime factors in common,

### The Fundamental Theorem of Arithmetic and Prime Factors

See the attached file. 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 Arithm

### Number Theory Perfect Square

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 Explanation

Write the numbers 25, 32, 56 to the base 5. write 47, 68, 127 to the base 2.

### RSA and Digital Signature assistance

RSA and Digital Signatures In a digital signature, scheme, there are two algorithms, sign and verify. The algorithm sign takes a secret key and a message, then outputs a signature. Please see attached for question

### 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:

### Asymptotes, Synthetic Division and Even and Odd Power Theorem

1. FIND ALL VERTICAL ASYMPTOTES OF THE FUNCTION. *******X+5 F (x) ------------------- ****4x squared+7 2. WHICH SHOWS THE TRUE STATEMENT FOR THE GRAPH OF THE RATIONAL FUNCTION g. *******X+2 g(x) ------------------- ****x squared+2x-3 3.USE SYNTHETIC DIVISION TO FIND UPPER AND LOWER BOUNDS OF THE REAL ZEROS OF f.

### 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

### Polynomials and Rational Functions : Synthetic Division

1.USE DESCARTES` RULE OF SIGNS TO DETERMINE THE POSSIBLE NUMBER OF POSITIVE REAL ZEROS OF THE FUNCTION. f(x)=3x cube-4x squared-2x-4 ANSWER: A.3 or 1 B.1 C.2 or 0 D.5, 3, OR 1 2.USE SYNTHETIC DIVISION TO COMPLETE THE INDICATED FACTORIZATION:x cube-7x-6=(x-3)( ) 3.USE SYNTHETIC DIVISION TO COMPLETE THE INDICA