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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.

Filip Haglund
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    I think that your question is equivalent on asking which is the smallest number we don't know if it is prime or composite. – Crostul Jan 21 '16 at 12:27
  • Yes, it is. I didn't quite know how to express the question. – Filip Haglund Jan 21 '16 at 12:29
  • $n>10^{128}$ since http://www.numberempire.com/primenumbers.php can check up to $10^{128}$. – wythagoras Jan 21 '16 at 12:29
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    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. – hardmath Jan 21 '16 at 12:38
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    @wythagoras But most of them haven't been checked yet. So we don't know of them whether they're prime or not. – Daniel Fischer Jan 21 '16 at 12:39
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    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? – David K Jan 21 '16 at 13:13
  • @DavidK a rough number is fine. We, as in humans, cats or any species we know of, including computers. – Filip Haglund Jan 21 '16 at 13:16
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    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. – DanaJ Jan 21 '16 at 13:27
  • Discussions of the Baillie-PSW Primality Test will often mention that it is known not to fail up to $2^{64} \approx 1.8e19$, which seems to imply an exhaustive knowledge of primes up to that limit. For details of what precisely supports that claim, see this explanation by Thomas Nicely. – hardmath Jan 21 '16 at 14:36
  • @FilipHaglund Even a rough number is in some sense unknowable; perhaps there's someone who has gotten 20% farther than anyone else but has not yet told anyone, and even they cannot be sure that someone else has not secretly gotten yet another 30% farther already. – David K Jan 21 '16 at 15:21
  • Another detail: if the number has been tested for primality, but the result isn't stored somewhere in some form, does it count as "known"? (The "some form" of storage could be a list of just primes in a compressed encoding; but it's not clear to me that "known" is satisfied by "knew it once and could figure it out again".) – David K Jan 21 '16 at 15:26
  • @hardmath, re BPSW, the claim is made because an exhaustive list of Fermat base-2 pseudoprimes is known, which required a combination of clever math to reduce the candidates plus years of computation. That enables testing a (relatively) small set of possible counterexamples. The test will always pass on primes. Nobody (to my knowledge of course) has run the test on every prime up to 2^64, and certainly nobody has run all numbers to 2^64. – DanaJ Jan 21 '16 at 16:14
  • Related to @hardmath's comment on BPSW, if we go with that argument, then we can use http://arxiv.org/abs/1509.00864 to claim the same for all n < 3317044064679887385961981. There exists a very fast deterministic test for primality in the range. Which then leads one to wonder why APR-CL or ECPP (or AKS) couldn't be used to claim the same for larger results, with no limit unless we arbitrarily decide on a time limit (e.g. in under 100 years with current software/hardware) – DanaJ Jan 21 '16 at 16:22
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    See also Number we know all prime numbers less than, which is closed as duplicate of that earlier one. However its Accepted Answer by Henning Makholm is worth reading. – hardmath Jan 21 '16 at 18:33

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