1) I can't really follow the Miller-Rabin Primality Test. What does it mean for something to be a pseudoprime for the base a? Why would we even want to know if something is prime? Why is that an advantage to us? Does composite mean factorable? Why is the success of these two methods not guaranteed?
2) I think it is fascinating that we have these different tests to test if something is prime or not. The fact that we can do it with such giant numbers is amazing, and that our computers can do it so quickly. I am still trying to figure out why this would be of so much use to us.
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