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Number Theory

Math 3200-001

Homework for Spring 2015

  1. Due F 1/16
    Exercise 1.1. Use Algorithm 1.1.13 and show all of your work.
  2. Due F 1/23
    Exercise 1.2. Use Algorithm 1.2.3 and show all of your work, displaying P and X at each step.
    Exercise 1.7. Also compare $\pi(y)$ to $\frac{y}{\log(y)-1}$ for $y=10000$. You may use a computer on this problem.
    Exercise 1.8. Feel free to use the Fundamental Theorem of Arithmetic on this problem.
  3. Due F 1/30
    Exercise 2.2. Use Algorithm 1.1.13 and show all of your work. Feel free to use python to do (and display) the computations, but in this case also turn in a printout of your code.
    Exercise 2.7.
    Exercise 2.8. Optional Challenge problem: Exercise 2.9
    Exercise 2.24.
  4. Due M 2/09 Show all computations
    Exercise 2.10.
    Exercise 2.11.
    Exercise 2.13.
    Exercise 2.18.
  5. Due M 2/16
    Exercise 2.23.
    Exercise 2.26.
    Exercise 2.30. Do this by hand, showing all your work.
  6. Due F 2/27
    Find all of the irreducible polynomials of degree $4$ over the field $\mathbb{Z}/2\mathbb{Z}$.
    Which of the polynomials that you found are primitive?
    Show and explain your work!
  7. Redo the midterm (correctly this time). You will present this corrected version to me in my office. Sign up for a time slot in class. When you come to my office bring both the original and the corrected versions with you.
  8. Due M 4/20
    Use the factor base method to factor the integers $9073$ and $9509$. For both integers $n$ use each of the following methods for generating the possible $B$-smooth integers:
    1. For $k=1,2,3,\dots\ \ $ try the integers $\lfloor\sqrt{kn}\rfloor$ and $\lceil\sqrt{kn}\rceil$.
    2. Try the numerators of the partial convergents in the continued fraction expansion of $\sqrt{n}$.
    Use the base $B=[-1,2,3,5,7,11,13,17,19]$.
  9. Due W 4/29
    1. For each prime $p=3,5,7,11,13$ consider the elliptic curve $E$ over $\mathbb{Z}/p\mathbb{Z}$ with equation $y^2=x^3-x$. Find the order and structure of $E$ as a product of cyclic groups.
    2. For each prime $p=5,7,11,13$ consider the elliptic curve $E$ over $\mathbb{Z}/p\mathbb{Z}$ with equation $y^2=x^3-1$. Find the order and structure of $E$ as a product of cyclic groups.
  10. Due F 5/1
    Factor the integers $9073$ and $9509$ (once again) this time using Lenstra's elliptic curve factorization method as discussed in class.

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