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In January this year the biggest prime number so far has been found and it is a Mersenne Prime. $$2^{57885161}-1$$.

My question: Are all prime numbers from 0 up to $2^{57885161}-1$ found, or not?
If you only look for prime numbers which are Mersenne Primes, you skip quite a few.

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No. In fact the number of prime numbers up to the largest found so far, by the prime number theorem, exceeds by a very large factor the number of estimated atoms in the universe. It would be prohibitive to enumerate them. – A. Webb Oct 7 '13 at 15:01
The testing for Mersenne primes is especially efficient, so there are certainly many unpublished primes below the largest known Mersenne prime. It's a bit insubstantial to try and pin down the range in which all primes have been "found", but there are not enough books to hold all that exist below the largest Mersenne prime you mention. – hardmath Oct 7 '13 at 15:02
up vote 6 down vote accepted

As you may know, GIMPS is conducting a search. They give each user a different prime number to test. Then they give that number to a different user to check.
They only test numbers of the form $2^p-1$, where $p$ is also prime. So they have only checked a few million possible values of $p$, that is a few million possible values of $2^p-1$.

The link shows that they have checked all primes $2^p-1$ if $p$ is below 44,576,437, and have double-checked primes $2^p-1$ for values of $p$ up to $p=26,187,517$.

There are around $(2^{57885161})/57885161$, or around $2^{57885135}$ unknown primes below the current record.

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I think your denominator should be adjusted for taking a natural logarithm, rather than log base 2. – hardmath Oct 7 '13 at 15:16
+1 but closer to $2^{57885136}$, actually (natural log versus log base 2). – Erick Wong Oct 7 '13 at 15:16
@Michael Yes, I am familiar with GIMPS (thanks to Numberphile) Thanks, I will have a look at their website. – Rik Oct 7 '13 at 15:28
Indeed, the exponent is actually 57885135.742... – Charles Oct 24 '13 at 16:45

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