The largest known prime number has emerged. It has more than 23M digits(washingtonpost.com)
washingtonpost.com
The largest known prime number has emerged. It has more than 23M digits
https://www.washingtonpost.com/national/health-science/the-largest-known-prime-number-has-emerged-it-has-more-than-23-million-digits/2018/01/12/e641cef0-f6e2-11e7-beb6-c8d48830c54d_story.html
35 comments
I wish we can collectively "mine" solutions for questions that are currently un-solved.
Like Larger prime number, or create a hive mind that acts as an AI.
Like Larger prime number, or create a hive mind that acts as an AI.
Isn't that the point of projects like folding@home?
And hive mind AI is literally skynet.
And hive mind AI is literally skynet.
The Borg would be a better analogy; Skynet was just an AI with a specific purpose, no hive-mind about it.
Well that's more reassuring.
There are attempts to do this, for instance Primecoin and Curecoin.
There's a site for large prime numbers, and I wondered Mersenne primes have been the largest primes known - since 1992, and since 1952 with an exception for 1989 to 1992. GIMPS - https://www.mersenne.org - has topped the list since 1996.
https://primes.utm.edu/notes/by_year.html#table2
https://primes.utm.edu/notes/by_year.html#table2
I have two questions:
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'?
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'?
1. Mersenne numbers (of the form 2^n - 1) are far easier to prove primality than other similarly sized numbers.
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
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
> Are all prime numbers smaller then this one also known?
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.
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.
1. Essentially, yes there's a list. Since there doesn't exist a higher level formula to identify primes it's been verified by some variation of a brute force method.
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.
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.
Calling it brute force isn't quite right. There is an efficient algorithm for prime determination (Lucas-Lehmer primality test) on Mersenne Primes (numbers of the form 2^p-1).
https://en.wikipedia.org/wiki/Lucas–Lehmer_primality_test
https://en.wikipedia.org/wiki/Lucas–Lehmer_primality_test
Thanks for the reminder! And this is my bad because I was paywalled on my phone and didn't see that it was, in fact, a mersenne prime that was found.
We know for sure that we don’t know all smaller primes.
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)
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)
What’s the significance of primes being one less than large powers of 2? Does this mean base-2 is the most natural base?
2 is the only integer n such that n-1 = 1.
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-...)
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-...)
Here's a page on the distribution of prime numbers.
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...
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...
https://www.mersenne.org/primes/
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.
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.
Previous discussion: https://news.ycombinator.com/item?id=16063946
Given that we know that there are infinitely many primes, why are we interested in knowing ever larger ones?
It's somewhat like asking, what is the largest known natural number?
It's somewhat like asking, what is the largest known natural number?
If nothing else, "what's the largest prime number you've found?" would be a fun question to ask another intelligent civilization, should we make contact with one.
Alright, that's a good one :)
Prime numbers thin out. We have very good estimates of how they thin out. We also know that gaps between primes can get arbitrarily large. And we cannot predict when the next prime will happen without huge computations. And large primes have a use, e.g. cryptography (although, who knows when we'll have a use for such a large one). Having a list of primes is also very useful for empirical evidence for many conjectures in number theory.
None of these things are true about natural numbers in general.
None of these things are true about natural numbers in general.
For cryptography it isn't very useful to know the largest one. Think about it ... Your other answers amount to: it is difficult, that's why we do it. ?! Sure, just don't expect me to care about it.
No one can predict the next prime. That's why people want to compute the next prime.
Because nobody knows a fast algorithm to factorize an integer into its prime factors. Yet, it is very easy to check if a set of prime factors multiply to a certain integer.
https://stackoverflow.com/questions/439870/why-are-primes-im...
https://stackoverflow.com/questions/439870/why-are-primes-im...
Innate curiosity isn't enough?
Quote from the Article: """The figure is calculated by multiplying 2 by itself 77,232,917 times and then subtracting 1."""
Would HN agree that multiplying 2 by itself once is 2⨯2 = pow(2,2), twice is 2⨯2⨯2 = pow(2,3) and N times yields pow(2,N+1)?
The Mersenne Prime found is pow(2,77232917) − 1, hence the article got the number wrong?
Would HN agree that multiplying 2 by itself once is 2⨯2 = pow(2,2), twice is 2⨯2⨯2 = pow(2,3) and N times yields pow(2,N+1)?
The Mersenne Prime found is pow(2,77232917) − 1, hence the article got the number wrong?
If you're going that way, you can also interpret it - pardon my programming - as the following:
int result;
for (int i = 0; i < 77232917; i++) {
result = 2 * 2;
}
return result - 1;
int result;
for (int i = 0; i < 77232917; i++) {
result = 2 * 2; <----- you basically said result=4 77232917 times
}
return result - 1; <--- and then 4-1=3
That is the joke.
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Yeah, I suppose if one wants to be pedantic. But we know what it means.
Dang ol' off-by-one errors.
Maybe mining zcash is lucrative, but finding a new big prime number sounds pretty nice