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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.

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    $\begingroup$ $$293703234068022590158723766104419463425709075574811762098588798217895728858676728151577$$ $\endgroup$ – Moo Jan 8 '18 at 23:11
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    $\begingroup$ 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. $\endgroup$ – Moo Jan 8 '18 at 23:14
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    $\begingroup$ Not even close! See: en.wikipedia.org/wiki/Largest_known_prime_number . It has 23 Million + digits! $\endgroup$ – Moo Jan 8 '18 at 23:19
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    $\begingroup$ 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: primes.utm.edu/notes/GapsTable.html (a very nice site on primes). $\endgroup$ – Moo Jan 8 '18 at 23:22
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    $\begingroup$ WolframAlpha can do this computation and it agrees with Moo: wolframalpha.com/input/… $\endgroup$ – Qiaochu Yuan Jan 9 '18 at 2:59

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