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We already know some very big prime numbers. ($2^{257,885,161} − 1$ as of time of writing is the largest known)

It is my understanding, that we know it is a prime number but we don't know all prime numbers smaller than it.

An equivalent of this is it is like if we know 7, 3 and 2 are prime numbers though we still haven't found 5.

My question is this, what is the largest prime number that we know all prime numbers smaller than it?

A link to a list of said prime numbers would also be heavily appreciated.

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marked as duplicate by muaddib, user147263, Ben Millwood, user91500, Claude Leibovici Jul 19 '15 at 4:26

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  • $\begingroup$ See, one problem with this is that whatever list you have, I can make a stream-based prime sieve algorithm, wait, and then give you one more prime than you had already. I can even save computing effort by hardcoding your list into the seed of my algorithm. $\endgroup$ – Ian Jul 19 '15 at 0:17
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It's not really a thing that's being kept track of.

It's easy to find downloadable lists of primes smaller than a few billion (e.g. at https://primes.utm.edu/lists/small/).

But if you go a few orders of magnitude above that -- say, if you want a list of all primes up to a trillion -- storing and distributing the lists themselves begin to be more tedious and expensive than just recomputing the primes when you need them. There are 37,607,912,018 primes less than a trillion; storing all of them would fill a medium-sized hard disk. And what for? Most of the things you could do with a table of such primes, such as looking up a number to see if it's prime, can be done with less resources simply by testing it for primality from scratch. (Testing whether a 12-digit number is prime is quick and simple if you have a computer, and if you don't have a computer how would you even store or access a table with 37 billion entries?).

So, depending on what you mean by "known", the number you're looking for probably changes from day to day, as someone somewhere computes a longer list of primes than anyone else was keeping yesterday, or deletes their list again to use their disk space for something more productive.

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