Homework 4

Due Tuesday, March 26

- Write a proof like Euclid's proof of $(x-y)(x+y) + y^2 = x^2$ for the identity $x^3 - y^3 = (x-y)(x^2 + xy + y^2)$
- What is the etymology of the word "algorithm"?
- Where did Euclid demonstrate the Euclidean algorithm (Which proposition in which book)? Where did he prove the fundamental theorem of arithmetic. If you can, describe his proof.
- Find a formula expressing the length of a side of a regular pentagon inscribed in the unit circle. (You can look it up if you must, but I'd rather you figure it out for yourself. I may provide a hint here.) The formula you come up with should involve a square root. Use it to write a Euclidean (straightedge and compass) construction of a regualar pentagon. If you do need to look it up, do so in Euclid, not some random place on the web.
- Find out how far the search for Fermat primes has gone (with, as we know, only negative results).
- What is the etymology of the word "cyclotomic"?
- Do Exercise 4.38 (page 199) in our text.
- Write the short essay discussing what you learned from our work on number theory. You don't need to summarize the material - I'm more interested in your response to it.
- Suggest several (three?) possible topics for your term paper/presentation. Each should be a paragraph or two showing why you want to do it and where you might begin reading.
- (Optional) Write a tail recursive implementation of the Euclidean algorithm.