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New to this, sorry for bad formatting but:

The largest know prime number is a Mersenne Prime Number, i.e. it is in the form $2^n-1$. Whilst this is big, the last known prime number before this was millions of digits shorter. Take the list $2, 3, 5, 7, 11,\cdots$ in the list $11$ is the largest (prime) number. We know all the prime numbers before $11$, though. So, a better way to ask would be as stated below:

What is the largest prime number where we know all the prime numbers before it? Doing some research, I've found that we know the first $50$ million primes or so, but I'm sure with all the computing power we humans have, that cannot be the limit.

Cryogenic
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  • Say what? $ $ $ $ – Kaster Dec 23 '16 at 21:45
  • Sometimes I think it pays to just use some part of the question as the title; in this case that would be, "What is the largest prime number where we know all the prime numbers before it?" That seems a perfectly clear statement of the question and I can't think of a better way to ask it in so many words. – David K Dec 23 '16 at 21:54
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    My guess is that it's much greater than $10^{100}$. Wolfram Alpha takes barely a couple of seconds to tell me that $10^{100} + 267$ is the next prime. My next query gave me the prime $10^{1000} + 453$. – Robert Soupe Dec 23 '16 at 22:01
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    @RobertSoupe: That would require a somewhat flexible definition of "known". All of the primes below $10^{100}$ cannot possibly be listed explicitly anywhere -- the observable universe contains too little matter to build enough storage media for that. – hmakholm left over Monica Dec 23 '16 at 22:08
  • How many prime numbers are known? is a near-duplicate of this. – hmakholm left over Monica Dec 23 '16 at 22:09
  • On the other hand, Wolfram Alpha couldn't give me an exact value for $\pi(10^{100})$. I dropped down to $\pi(10^{10})$ to get the answer 455052511. This suggests that with the right programming, you can have your computer make you a list of the first 400 or 500 million primes. It might take up a gigabyte on your hard drive (something I didn't think about until I saw @Henning's comment). – Robert Soupe Dec 23 '16 at 22:10
  • Upon further reflection, my NextPrime queries aren't all that informative. The powers of 10 that I used happened to be in relatively small prime gaps, so even by crude trial and error, it would not take too much time to find the primes I mentioned earlier. – Robert Soupe Dec 23 '16 at 22:14
  • @Henning Yeah, that's the duplicate alright. – Robert Soupe Dec 23 '16 at 22:16
  • @HenningMakholm: Thanks for the duplicate checking, unfortunately I didn't find any of those during my (by your standards, brief) searching. – Cryogenic Dec 23 '16 at 22:34
  • @RobertSoupe Wolfram Alpha is a great mathematical engine, though most of the questions you asked are simply stored in a massive database ready to be fetched later, the exponent of pi is one of millions of examples. – Cryogenic Dec 23 '16 at 22:34
  • @DavidK obviously so. I'm really bad at wording my questions but thank you for the point-out. – Cryogenic Dec 23 '16 at 22:35
  • @Cryogenic: I used the advanced technique of skimming through the "Related" links shown to the right of this comment thread, and following the duplicate links from there. :-) – hmakholm left over Monica Dec 23 '16 at 22:38
  • @HenningMakholm I see :p. Might use it later lol – Cryogenic Dec 23 '16 at 22:38
  • @Cryogenic Fair enough. Though I've just now tried $\pi(10^{10} - 3^8)$ and it seems to have given me the answer just as quickly. The speed of my Internet connection would probably be the only difference between Wolfram Alpha and if I had my own copy of Mathematica at home. – Robert Soupe Dec 24 '16 at 03:17

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