Section12.7Exercises

  1. Check that \(1729\) and \(2821\) are also Carmichael numbers.
  2. Find a Carmichael of the form \(7\cdot 23 \cdot p\) for a prime \(p\).
  3. Use either the Fermat or Mersenne coprime fact to provide a different proof that there are infinitely many primes.
  4. Pick some 4-6 digit numbers that don't share a factor with \(30030=2\cdot 3\cdot 5\cdot 7\cdot 11\cdot 13\). Find factors by trial division.
  5. Do the same with Fermat Factorization. Try to create a number that takes five steps with Fermat and with trial division.
  6. Try using Pollard Rho on a large number you create out of a few big primes (not too big!) with different seeds. Can you get it to take longer than a few turns? Get your prize numbers; now try factoring again with this method where you have changed the polynomial to \(x^3+1\) or something else other than \(x^2+1\).