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Where do the first gaps start appearing in what numbers we know are prime or not? For example this website http://compoasso.free.fr/primelistweb/page/prime/liste_online_en.php has a list of all the primes up to 1 trillion. How much higher could you go before you reach territory where it's unknown which numbers are primes and which aren't?

Samantha Clark
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  • We know that there are an infinite number of primes and we also know that there is an upper bound between two primes. From this we can conclude that there will always be prime numbers, no matter how large the numbers get. – David Jun 30 '17 at 19:14
  • Well I mean we know all of the primes between 1 and 100 with, but at what point do we become unsure of which numbers are prime and which aren't – Samantha Clark Jun 30 '17 at 19:20
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    There are fairly efficient algorithms for determining whether a number with thousands to millions of digits is prime or not (and even more efficient algorithms to give an answer with a very low probability of a false positive, and no possibility of a false negative). At that point, there are more primes below that to list than would be practical to print out. – Daniel Schepler Jun 30 '17 at 19:34
  • What kind of an answer do you want? Mathematica running on my laptop can tell which of the numbers in the range $10^{150}+n,1\le n\le2000$ are primes in a split second: $$n=67, 427, 771, 1323, 1443, 1459, 1753, 1911$$ are, the rest are not. Testing whether a given integer $m$ is a prime or not is quite fast for quite large $m$. But, it is pointless to build a list of all the primes up to that range (for the usual reason that the list would fill up the known universe in print). – Jyrki Lahtonen Jun 30 '17 at 19:35
  • Well yeah that's kind of what I'm curious about, if I were to try and find a list of prime numbers what's the largest list I could find or has ever been made – Samantha Clark Jun 30 '17 at 19:38
  • @SamanthaClark One can easily make a list much longer than 1 trillion. It's just that lists of that size tend to... become quite large. – Eff Jun 30 '17 at 19:40
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    @David: "we also know that there is an upper bound between two primes." What do you mean? The set of gaps between consecutive primes is unbounded, by an elementary trick using factorials (consider the numbers from $n!+2$ to $n!+n$ for arbitrarily large $n$). – symplectomorphic Jun 30 '17 at 20:44
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    I don't think anyone is keeping track of that. It's been asked here. https://math.stackexchange.com/questions/1366044/number-we-know-all-prime-numbers-less-than. Frankly, although I think this is a mostly unknowable questions, I think some of the "It's a nonsense question" are rude. I don't know what the answer is. – fleablood Jun 30 '17 at 21:06
  • https://primes.utm.edu/notes/faq/LongestList.html According to this it's about $10^{18}$. – fleablood Jun 30 '17 at 21:09
  • Used to be 4e18, but another project is extending it and will hit 5e18 in a month or so, and it will continue to be extended. Even that gets into debates about whether all the primes were really calculated since it skips swaths smaller than the minimum search gap. The "largest list" is hard to answer since we can trivially extend any list, and we can generate them fast enough to make it a meaningless project regardless. For the frontier of knowability, we'd be in the 30000+ digit range, as that's the limit of current practical general form proofs. Almost-certainly-prime much larger. – DanaJ Jun 30 '17 at 21:16
  • I think it's weird that this was closed as "unclear" - it's perfectly clear what's being asked, and I think Daniel Schepler's comment is a good answer to it. I've upvoted and voted to reopen. – Noah Schweber Jun 30 '17 at 21:57
  • @NoahSchweber The question has asked many times on various SE forms, on SO, and on other sites. The answer is complicated and turns into a philosophical quesiton about what "knowing" means. It needs to be much narrower to get a practical answer. – DanaJ Jun 30 '17 at 23:34
  • I am glad this question was incorrectly closed as "unclear what you're asking" instead of as a duplicate. Mwahahahahahahahahahahahaha! – The Short One Jul 01 '17 at 21:22

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