Homework 3

Homework 3

Due Tuesday, March 5

  1. Write a two or three page essay synthesizing what you learned about mathematics and the history of mathematics from our work on the parallel postulate. Please make this interesting for me to read. Don't just summarize topics. Consider telling me what was hard, what was easy, what was fun, what you would have liked more or less of, how the material connects to what you knew and to what you hope to know. Write your document in $\LaTeX$. Proofread (or have a friend proofread) for spelling and grammar. If two of you want to write a joint essay, structure it as a dialog.
  2. Show that if $2^n -1$ is prime then $n$ itself must be prime. (This is Exercise 4.3 in the text, There's a hint there if you need it.)
  3. Show that the converse to that statement is false.
  4. What is the largest known prime?
  5. What is the largest known non-Mersenne prime?
  6. Show that $2^{n-1}(2^n - 1)$ is perfect when $2^n -1$ is prime. Hint. Write down all the divisors of 496 and look at a natural way group them in order to add them up. Then generalize.
  7. Use the graph cskxReH.png to make some predictions for future largest known primes. Note that the y axis represents the number of digits of prime $p$ (that is, the logarithm of $p$) and that it's a logarithmic scale. That function grows very quickly! What's a formula for it? Note: if you read ahead in the article containing this image you will find answers to some of these questions. [ http://primes.utm.edu/notes/by_year.html ]
  8. On page 172 our text mentions the Pythagorean triple $(12709, 13500, 18541)$ known to the Babylonians. Show how to construct it using our characterization of primitive Pythagorean triples as $(m^2 - n^2, 2mn, m^2+n^2)$.
  9. Show using Euler's method that the Diophantine equation $x^4 - y^4 = z^2$ has no nontrivial solutions. (This is Exercise 4.21 in the text.)
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