Homework 3

Due: Sep 22nd, 2021

Term paper

Congratulations on choosing a topic for your term paper! The next step is to write an outline.

Instructions

Please find and use the LaTeX template for homework.

Use LaTeX to create a PDF. Upload your PDF to Gradescope. If you don’t have LaTeX on your computer, you can use Overleaf. Don’t submit the LaTeX source, just the PDF. Email your instructor (that’s me) if you have questions or need help.

Please include your name and the homework number within the document. Some additional formatting instructions are in the syllabus. To summarize:

  • Use a new page (\newpage) for each problem.
  • State which question you are answering and the actual question. Then, start your answer in a new paragraph.
  • Use environments such as proof and theorem (via \begin{proof}...\end{proof}).
  • Use 12pt option \documentclass[12pt]{amsart} and linespacing \linespread{2.4}.

You are encouraged to work together on the homework!

Problems to turn in on Gradescope

  1. Exercise 1.4.1.

  2. Exercise 1.4.3.

  3. Section 1.4, Exercises 8, 9, 10, 16: Try as many as you can. Turn in the best/most interesting one that you are able to solve.

  4. Here are three exercises. Turn in the best/most interesting one that you are able to solve.

    1. Exercise 1.4.11.

    2. Exercise 1.4.12. (This is challenging.)

    3. (Not from the textbook.)
      Let \(f_n\) be the \(n\)th Fibonacci number, defined by \(f_0=0\), \(f_1=1\), \(f_{n+1} = f_n + f_{n-1}\) for \(n \geq 1.\) Starting from \(f_0\), the sequence is \(0,1,1,2,3,5,8,13,21,34,\dotsc\). Let \(F_n\) be the polynomial \(F_n = x^{f_n}-1 \in \Bbbk[x]\). For example, \(f_5=5\), so \(F_5 = x^5-1\).

      • Find \(f_{15}\), \(f_{25}\), and \(\gcd(f_{15},f_{25})\). Also find \(\gcd(15,25)\).
      • Find \(F_{15}\), \(F_{25}\), and \(\gcd(F_{15},F_{25})\).
      • What is \(\gcd(F_n,F_m)\)? Give a clear statement.
        You might want to test your statement for more values of \(n\) and \(m\).
      • Optional: Try to prove your statement for \(\gcd(F_n,F_m)\).

Problems to complete on WebWork

Visit WeBWork. Complete the “Section 1.5” assignment within WebWork.

Additional reading, required

  1. Start reading Chapter 2 of the textbook.

Additional reading, optional