We know some pretty large primes thanks to projects like GIMPS, but for most of the large primes we know there are undiscovered primes inbetween those primes. To what number are we certain we know of all prime numbers within that range.
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1we know things for certain up to about $1.8361 \cdot 10^{19},$ see the table of maximal gaps at https://en.wikipedia.org/wiki/Prime_gap#Numerical_results – Will Jagy Jan 29 '19 at 22:30
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This has been asked multiple times. See these. https://math.stackexchange.com/questions/330142/currently-what-is-the-largest-publicly-known-prime-number-such-that-all-prime-n, https://math.stackexchange.com/questions/1851936/how-far-is-the-list-of-known-primes-known-to-be-complete – Gnumbertester Jan 29 '19 at 22:31
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This "smallest" number cannot be determined.
The reason is the following :
We cannot list all the gaps for the primes, upto , lets say $10^{200}$. So, there must be a prime $p<10^{200}$, for which the next prime $q$ is unknown. But it is easy to calculate this prime $q$ (even if we only allow proven primes), contradicting the existence of such an example. The problem is that it is "too easy" to find "small" primes.
This is also discussed in Chris Caldwell's prime page. A similar question is answered there - What is the smallest number which is unknown to be prime or not ?
If you mean an upper bound of the gap , Will Jagy's comment is a good answer to your question.
Peter
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