- Check that \(1729\) and \(2821\) are also Carmichael numbers.
- Find a Carmichael of the form \(7\cdot 23 \cdot p\) for a prime \(p\).
- Use either the Fermat or Mersenne coprime fact to provide a different proof that there are infinitely many primes.
- 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.
- Do the same with Fermat Factorization. Try to create a number that takes five steps with Fermat and with trial division.
- 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 \(x^2+1\).
