Wrapup Homework

This is the last homework assignment for the course. It's due Tuesday, May 21 (one week after the last class). Submit a pdf attached to email to my gmail address: moc.liamg|reklobe#moc.liamg|reklobe.

The first question calls for a short essay. The ones after that are based on the class presentations. You may not be able to answer them all - but you should at least be able to write good solutions for the ones based on your final paper.

(1) Write a short essay summarizing your experience in this course. Here are some questions you can use to organize your essay (but you need not answer each of them separately, or at all). What did you find most (least) interesting? Why? What was hardest (easiest)? (That may or may not match what was most interesting.) What are you most likely to remember (or forget)? How (if at all) did the work we did connect to what you study in your "regular" math courses? When were you bored or confused? What should I do differently if I teach the course again. What other questions ought I ask here? Finally, if you're willing: what grade do you think you've earned (base your answer on the amount of work you did and the amount you learned that was new, not the fraction of the course that you'd be willing to be tested on).

(2) Set theory (Rob's presentation). Zermelo's Axiom III (page 92 in our text) says, essentially, that whenever you can state a logical assertion about an object then there is a set whose elements are just the objects for which the assertion is true. For example, since you know that a number is even if and only if it is divisible by 2, you are allowed to talk about the set of all even numbers. Use Axiom III to prove that there is an empty set. (Hint: Can you think of a statement about an object $x$ that can't ever be true?). Then use Axiom III to prove that if $a$ is an object there is a set $A$ whose only element is $a$ (Hint: find the right assertion.) (This is essentially Exercise 2.30 in the text.)

(2a) (Optional) Set theory (Rob's presentation). Read something about Hilbert's Hotel in at least two places on the web. (One "reading" can be watching a youtube video). Report on what you discovered/liked/disliked. Can you show how Hilbert's Hotel can accommodate infintely many busloads (1, 2, 3, …) each of which carries infinitely many new guests (1, 2, 3, …)? Hint: Put guest $i$ from bus $j$ in room $p^j$, where $p$ is the $i^{th}$ prime. Which rooms remain empty then? How does this argument depend on something Euclid proved?

(2b) (Optional) Set theory (Rob's presentation). Spend a little time reading Calkin and Wilf's article Recounting the rationals,
Amer. Math. Monthly 107 (2000), p 360, also available at http://www.math.upenn.edu/~wilf/website/recounting.pdf. How does this relate to Rob's presentation? Tell me something interesting you found there.

(3) Chinese Remainder Theorem. Solve the simultaneous congruences

(1)
\begin{align} x \equiv 2 (\mod 8) \\ x \equiv 3 (\mod 13) \\ x \equiv 5 (\mod 21) \end{align}

How did I choose the six numbers in this question? Can you formulate a generalization?

(4) Chinese Remainder Theorem. Find an example of a pair of congruences that do not have a simultaneous solution (because the moduli are not relatively prime).

(5) Chinese Remainder Theorem.

An old woman goes to market and a horse steps on her basket and crashes the eggs. The rider offers to pay for the damages and asks her how many eggs she had brought. She does not remember the exact number, but when she had taken them out two at a time, there was one egg left. The same happened when she picked them out three, four, five, and six at a time, but when she took them seven at a time they came out even. What is the smallest number of eggs she could have had?

(You can find this quote lots of places on the web - the solution is there too. Please do it yourself!)

(6) Euler's conjecture. Elkies discovered that

(2)
$$2682440^4 + 15365639^4 + 18796760^4 = 20615673^4.$$

Show that this equation is correct $\mod 10$. Show that it's correct $\mod 9$ (Hint: the well known test for divisibility $\mod 9$ really computes the remainder). Show that the left and right sides of Equation (2) have the same number of digits (Hint: think about logarithms - don't just find a piece of software that will do the exact calculations for you.) Optional: Use the fact that $7 \times 11 \times 13 = 1001$ to invent a good algorithm for computing remainders $\mod 1001$ for a number expressed in base $10$, and use your algorithm to check Equation (2) $\mod 1001$. Now you know it's correct modulo 2, 3, 5, 7, 11 and 13. That's a pretty good evidence.

(7) (Calendar) Find the date (in the current calendar) for the start of the Mayan long count that ended last December. You will have to make several assumptions to get this right. There may not even be a "right answer". Document your work, and show that you're close by finding an "answer" on the web.

(8) (Hyperbolic geometry) Let $\Gamma$ be a circle of Euclidean radius $r < 1$ centered at the origin.

1. Find the hyperbolic diameter $d$ of $\Gamma$. (Hint: this is a problem from a previous homework!)
2. Find the hyperbolic circumference $c$ of $\Gamma$.
3. In Euclidean geometry, $c/d$ is the constant $\pi$. What can you say about "$\pi$" in hyperbolic geometry?

(9) (Gamma function) Show that

(3)
\begin{align} \int_0^1 (-\ln(t))^n dt = \int_0^\infty t^ne^{-t} dt . \end{align}

(this is a calculus 2 exercise). That function is written $\Gamma(n+1)$. Show that

(4)
\begin{align} \Gamma(1/2) = \sqrt{\pi} . \end{align}

(You can look up proofs on the web - there are lots - as long as when you write them out for homework you make it clear that you understand what you've written.)

(10) Pythagorean triples.Show that there is just one value of z for which $(1000,y,z)$ is a primitive Pythagorean triple.

(11) Find the smallest and largest values of z for which $(1680, y, z)$ is a primitive Pythagorean triple.