Section22.4Exercises

  1. Find an arithmetic progression of length five with less than ten between primes.
  2. Find an arithmetic progression of length six or seven starting at a number less than ten.
  3. Prove that there can be only one set of “triple primes” – that is, three consecutive odd primes.
  4. Find the value of \(23\#\).
  5. Show that \(\left(1-\frac{2}{p}\right)=\left(1-\frac{1}{(p-1)^2}\right)\left(1-\frac{1}{p}\right)^2\).
  6. Let \(D(N)=\prod_{p<N}\left(1-\frac{1}{p}\right)\). Compute \(D(N)\) by hand for all \(N\) between 10 and 20, without adding the fractions (just “FOIL” it out). What patterns do you notice in the denominators? The numerators?
  7. Search a good book or the internet for an amazing fact about primes. Describe it in a way your classmates (or peers, if you're not in a course) will understand.