1: How do we know it's the largest known prime number? What qualifies as a 'known' prime number? Is there an official registry somewhere?
2: Are all prime numbers smaller then this one also known? Or could somebody still come up with an unknown 'not very large prime number'?
It looks like there is a registry! Details for adding an item are at https://primes.utm.edu/notes/faq/add_prime.html
2. We haven't enumerated the vast majority of the primes under this. There could even be mersenne primes as yet unexamined smaller than this, as the search is distributed. And whenever an ssl certificate is generated, two (hopefully) previously undiscovered prime numbers is found. (For 1024-bit length keys, that's around 308 digits.)
The magnitude of the numbers involved is hard to think about. The number of protons in the observable universe "only" has around 80 digits.
References:
* Mersenne primes - https://en.wikipedia.org/wiki/Mersenne_prime
* Eddington number -https://en.wikipedia.org/wiki/Eddington_number
No, and they will never all be known.
There are approximately n/log(n) primes less than n. In this case n is 2^77232917 - 1. n/log(n) is approximately 8.6 x 10^2317930.
If you gave every person on Earth a million million processors [1], running at 1000 GHz each, that could find a prime each clock cycle, and each person used a time machine to send their processors back to when the dinosaurs went extinct so they by now they have all been running 65 million years...we'd have found about 1.4 x 10^49 primes.
Heck, if every particle in the universe was somehow made to computer, each of which could produce a new prime in the time it takes light to travel the Planck length (the smallest meaningful length possible in the universe), and they had all been doing this since the beginning of the universe, they would have discovered by now about 10^147 primes.
That's a mere 0.000...00012 percent of the way there, where I've skipped more than 23000000 zeros.
2. That all smaller primes are known can't be verified. Primes of this size are usually found by a number of formulas that find "likely" primes and then verified by brute force (afaik). Because of that fuzzy match to start there might be smaller primes that haven't yet been identified.
For example, there is at least one prime between 2^77,232,915 and 2^77,232,916 that we don’t know (https://en.wikipedia.org/wiki/Bertrand%27s_postulate)
(That’s a very loose bound. I think that, for any N > 2, there are more primes than squares of integers less than it)
That’s significant because (n^m) - 1 is divisible by n - 1.
So, for example, 125320078^35785332 - 1 is divisible by 125320077 and, hence, not a prime.
So, this problem has been solved for all n>2.
n^m + 1 is a more difficult problem. For example, for n=10, we get 2: prime, 11: prime, 101: prime, 1001: composite, 10001: composite, etc (at least the 8,000 following are composite. See https://math.stackexchange.com/questions/2108085/is-there-a-...)
http://primes.utm.edu/howmany.html
I tried to estimate how many prime numbers there were below this number, and Wolfram Alpha timed out, though I got the density estimated, I think.[1] If I remember my logarithm math, around one in 50 million numbers at this magnitude is prime, but that's still gigantic! The count of prime numbers smaller than this prime number is also around 23 million digits long.
[1] http://m.wolframalpha.com/input/?i=%28log+2%29+*+77%2C232%2C...
For Mersennes at least, which right now are the largest primes.
A prime is known once you've shown it is prime, i.e. no factors other than 1 and itself.