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So I know there are infinitely many prime numbers, for Euclid proved there were in $300$ BCE.

However, I cannot find the following prime number after

$293703234068022590158723766104419463425709075574811762098588798217895728858676728143227$

Let this prime number be $X$, then $X$ is $87$ digits long, and I want to find the following prime number from this but it is too far away! I am trying to find pairs of prime numbers with large gaps. For example, the following prime number after $p$ is $p + 972$ such that $p$ is equal to

$5748393059677584745738967520671906740165478593679375688574891672896728975389$

This is $76$ digits long, which is pretty amazing, but the point is, does anyone know what is the next prime number after $X$?

All I know is that if we let the next prime number after $X$ be $Y$, then $Y > X + 1510$.

Thank you in advance.

Mr Pie
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    $$293703234068022590158723766104419463425709075574811762098588798217895728858676728151577$$ – Moo Jan 08 '18 at 23:11
  • @Moo how did you find that out? What makes you so sure it is a prime number? – Mr Pie Jan 08 '18 at 23:13
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    I used a probabilistic primality test and verified that with a tool that can generate the next prime in sequence to validate my suspicions. The gap is 8350 from the number you wrote. – Moo Jan 08 '18 at 23:14
  • @Moo Well all the digits are the same except for the last $5$, and so we now find the answer to $51577 - 43227$ which is … $8350$ !!! WOW!!! Is that a new record???? Oh, I didn’t realise you edited that into your comment. – Mr Pie Jan 08 '18 at 23:17
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    Not even close! See: https://en.wikipedia.org/wiki/Largest_known_prime_number . It has 23 Million + digits! – Moo Jan 08 '18 at 23:19
  • Oh no I don’t mean a record concerning prime numbers. I mean a record concerning prime gaps! Sorry if I was misleading. – Mr Pie Jan 08 '18 at 23:21
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    I highly doubt it. I can choose a prime 10x larger than these and will likely find a much larger gap. Keep doing this forever and the gaps become larger. For example, see: https://primes.utm.edu/notes/GapsTable.html (a very nice site on primes). – Moo Jan 08 '18 at 23:22
  • Thank you very much @Moo. I hope all the best for you :) (surprisingly that rhymes) – Mr Pie Jan 08 '18 at 23:25
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    WolframAlpha can do this computation and it agrees with Moo: http://www.wolframalpha.com/input/?i=next+prime+after+293703234068022590158723766104419463425709075574811762098588798217895728858676728143227 – Qiaochu Yuan Jan 09 '18 at 02:59
  • Thank you @QiaochuYuan :) Now I don’t need to code a program – Mr Pie Jan 09 '18 at 03:39
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    It is a prime gap of maximum known merit: M=41.93878373153988 http://www.trnicely.net/#MaxMerit – pietfermat Jan 09 '18 at 07:48
  • Oh my... I knew of maximal gaps, but never had I heard of anything else relating to the matter, including merits. I did not intend on copyrighting anything... if I did...thank you for letting me know of this, so I am aware of upcoming progress of this sort some time in the future. – Mr Pie Jan 09 '18 at 10:02

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