How many prime numbers are known?

Wikipedia says that the largest known prime number is $2^{43,112,609}-1$ and it has 12,978,189 digits. I keep running into this question/answer over and over, but I haven’t been able to find how many known prime numbers exist. The website primes.utm.edu allows downloading of the first 50,000,000 known primes so I know there are at least that many; I’m not expecting to find a list of all known primes, but is there any information on how many there are known?

edit Relevant video from Khan Academy: Prime Number Theorem: the density of primes

Answer

Nobody’s really keeping count.

Newly discovered large primes make the news, but primes in the range of, say, a few hundred digits are not something that anybody keeps track of. They are very easy to find — the computer that’s showing you this text is likely capable of finding at least several ones per second for you, and with overwhelming probability they will be primes nobody else have ever seen before.

There are very many hundred-digit primes to find. We could cover the Earth in harddisks full of distinct hundred-digit primes to a height of hundreds of meters, without even making a dent in the supply of hundred-digit primes.

This also raises the question of what it means that a prime is “known”. If I generate a dozen hundred-digit primes and they are forgotten after I close the window showing them, are these primes still “known”? If instead I print out one of them and save the copy in a safe without showing it to anybody, is that prime “known”? What if I cast it into the concrete foundation for my new house?

Attribution
Source : Link , Question Author : f.ardelian , Answer Author : hmakholm left over Monica

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