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# Number Theory

### Complete Residue System : Suppose that p is a prime number

Suppose that p is a prime number. Show that 0, (p-1)!/2, (p-1)!/3, ..., (p-1)!/(p-1) is a complete residue system modulo p.

### One to One Map : Suppose that p is a prime number

Suppose that p is a prime number &#8805; 3 and r1, r2...rp is a complete residue system of modulo p. Prove that r1+r2+...+rp is divisible by p.

### Euclidean algorithm, primes and unique factorization, congruence

Topics usually include the Euclidean algorithm, primes and unique factorization, congruences, Chinese Remainder Theorem, Hensel's Lemma, Diophantine equations, arithmetic in polynomial rings, primitive roots, quadratic reciprocity and quadratic fields. (See attached file for full problem description)

### Polynomials, Integers and Divisibility

Problem 6. Suppose that f(s) = adxd+ad-1xd... is a polynomial with integral coefficients (so a0, a1 . . ,ad E Z). Show that f(n)?f(m) is divisible by n ? m for all distinct integers n and m. See the attached file.

### Infinitely Many Primes Using Euclid's Proof

Prove that there are infinitely many primes of the form 4n ? 1. (Hint: Modify Euclids proof that there are infinitely many primes. First, prove that any number of the form 4n ? 1 has a prime factor which is of the form 4k ? 1). See the attached file.

### Prime Factorization : Proof - Square-Free Integers

Show that every positive integer n can be written in the form n = ab where a is square-free and b is a square. Show that b is thi the largest square dividing n. (A square-free integer is an integer that is not divisible by any square > 1).

### Find Prime Factorization (by hand) of 2006 and 2007.

Find Prime factorization (by hand) of 2006 and 2007.

### Taylor Polynomials : Degrees and Plotting

2. (Taylor polynomials) (a) Write down the Taylor polynomials Pn(x) of degree n = 0, 1, 2,3 for the function f(x) = ln x about the point x = 1. (h) Plot the polynomials Pn(x) and the function f(x) on the interval [0, 3] using Matlab. Describe how the error |f(x) ? Pn(x)| behaves with respect to the point x and the degree n.

### Least common multiple

Find the least common multiple of 2345 and 5236, 10000 and 10001.

### Cryptography

List 4 problems dealt with by cryptography & give real world examples of each. 2 paragraphs please.

### Bezout's Theorem for Polynomials

(See attached file for full problem description) --- Suppose that a(x) and m(x) are relatively prime polynomials in F[x]. Bézout's Theorem guarantees the existence of polynomials u(x) and v(x) in F[x] such that... --- (See attached file for full problem description)

### 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,