What's the biggest $n$ such that for all $1<x<n$, we know for sure if $x$ is prime?

The smallest primes are easy to find, and the biggest ones we haven't found yet. At the top, we have Mersenne primes, but not all primes are Mersenne primes. There's an unclear boundry, up to which we know some of the primes, but probably not all.


marked as duplicate by hardmath, SchrodingersCat, Community Jan 21 '16 at 19:06

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    $\begingroup$ I think that your question is equivalent on asking which is the smallest number we don't know if it is prime or composite. $\endgroup$ – Crostul Jan 21 '16 at 12:27
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    $\begingroup$ So many numbers, so little time. There is no clear "boundary". The numbers that potentially can be checked far outnumber those that someone actually will check during the lifetime of the Sun and Earth. $\endgroup$ – hardmath Jan 21 '16 at 12:38
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    $\begingroup$ @wythagoras But most of them haven't been checked yet. So we don't know of them whether they're prime or not. $\endgroup$ – Daniel Fischer Jan 21 '16 at 12:39
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    $\begingroup$ Who are "we"? Suppose all numbers $2$ through $k-2$ have been checked by someone in the world at some point, $k$ has never been checked, and last year someone somewhere privately checked $k-1$ but never told anyone else the result. Does that make $k$ the largest $n$? If not, what is the standard of when "we" know whether a number is prime? $\endgroup$ – David K Jan 21 '16 at 13:13
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    $\begingroup$ The largest exhaustive search I know ended at 4e18 (Tomás Oliveira e Silva). The problem in some sense is that it takes only a couple seconds to get the next 1 million primes, so it's trivial to push the boundary forward. But finding all primes in the range 4e18 to 5e18 is an extremely time consuming task just because of the size of the range. You could also answer this based on a small time limit for primality testing any number less than n, which I think some others have implied. Depending on your time limit, n could be huge. $\endgroup$ – DanaJ Jan 21 '16 at 13:27

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