If we used such a machine to to enumerate Pi what would happen? Is it proven that Pi infinitely long? If so that would be a countable infinite because it is digits of precision, right?

Could we write algorithms that search all the digits of pi for patterns? Or mayb messages on speculation of creators or others who might be able to tamper with it?

Could other math things be searched? Have we proven there is no upper limit to prime numbers? I know the largest known mersenne prime fills a novel sized book with just one number, but when does a number get too big to be prime if ever?