Homework 4

Due Tuesday, March 26

  1. Write a proof like Euclid's proof of $(x-y)(x+y) + y^2 = x^2$ for the identity $x^3 - y^3 = (x-y)(x^2 + xy + y^2)$
  2. What is the etymology of the word "algorithm"?
  3. 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.
  4. 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.
  5. Find out how far the search for Fermat primes has gone (with, as we know, only negative results).
  6. What is the etymology of the word "cyclotomic"?
  7. Do Exercise 4.38 (page 199) in our text.
  8. 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.
  9. 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.
  10. (Optional) Write a tail recursive implementation of the Euclidean algorithm.
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