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?
The largest known prime number has emerged. It has more than 23M digits
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Re: The largest known prime number has emerged. It has more than 23M digits
#12Given 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?
No one can predict the next prime. That's why people want to compute the next prime.
Re: The largest known prime number has emerged. It has more than 23M digits
#13Quote 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?
Dang ol' off-by-one errors.
Re: The largest known prime number has emerged. It has more than 23M digits
#14Earlier quoted context omitted.
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.
Isn't that the point of projects like folding@home? 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.
Re: The largest known prime number has emerged. It has more than 23M digits
#15Given 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?
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.
Re: The largest known prime number has emerged. It has more than 23M digits
#16Given 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?
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...
Re: The largest known prime number has emerged. It has more than 23M digits
#17I 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. 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
Re: The largest known prime number has emerged. It has more than 23M digits
#18I 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'?
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...
Re: The largest known prime number has emerged. It has more than 23M digits
#19Given 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?
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…
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.
Re: The largest known prime number has emerged. It has more than 23M digits
#20Given 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?
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.