# Largest Known Prime

## Main Question or Discussion Point

This news is a little dated, but I still found it interesting and wanted to see what everyone else thought about this years discovery of a new "largest" prime: $2^{(57,885,161)}-1$ its 17,425,170 digits long and would span all 7 harry potter books twice. Written out in plain text it would take up 22.5mb!

Is the size of primes we find only going to be limited by our computing power? Is there any other way of finding mega primes that aren't Mersenne primes($2^p-1$)?

(Fun facts credited to Adam Spencer from his fascinating TED talk which can be found here: Adam Spencer: Why I fell in love with monster prime numbers #TED : http://on.ted.com/gmnG)

Borek
Mentor
its 17,425,170 digits long and (..) Written out in plain text it would take up 22.5mb!
And not 17.5 MB?

AlephZero
Homework Helper
its 17,425,170 digits long and would span all 7 harry potter books twice. Written out in plain text it would take up 22.5mb!
And not 17.5 MB?
You forgot about the commas after every three digits

Borek
Mentor
You forgot about the commas after every three digits
$$\frac 4 3 \times 17425170 = 23233560$$

Now 22.5 MB is 0.7 MB short.

Hey, I was just posting what I read and learned from that talk, lol.

pbuk
Gold Member
$$\frac 4 3 \times 17425170 = 23233560$$

Now 22.5 MB is 0.7 MB short.
FAIL $\frac{223233560}{2^{20}} = 22.5$

pbuk
Gold Member
Interestingly, its exactly 22.5 MB.

Borek
Mentor
Depends on the MB. But OK, if you select two arbitrary conventions, you can get this result.

I guess with two arbitrary conventions you can get ANY result :tongue2:

This news is a little dated, but I still found it interesting and wanted to see what everyone else thought about this years discovery of a new "largest" prime: $2^{(57,885,161)}-1$ its 17,425,170 digits long and would span all 7 harry potter books twice. Written out in plain text it would take up 22.5mb!

Is the size of primes we find only going to be limited by our computing power? Is there any other way of finding mega primes that aren't Mersenne primes($2^p-1$)?

(Fun facts credited to Adam Spencer from his fascinating TED talk which can be found here: Adam Spencer: Why I fell in love with monster prime numbers #TED : http://on.ted.com/gmnG)
Enjoyed video. Thanks. Looks like they are limited by our computing power and although there are methods to finding other primes, Mersenne primes take less effort to do so. You can google Primes, Mersennine primes, and the Lucas-Lehmer primiality test if you wish to learn more about it.

pbuk
Gold Member
Save time googling, start here.

I got a better idea. Since Jesse brought it up, how about he implement the Lucas-Lehmer algorithm in say Mathematica, on a number $2^p-1$ that he believes can be tested for primality in 9 hours, set it running before bed time say 10:00p, run it all night till 6:00a, and then report the results for us here.

Ok, little more work for you Jesse. Got time? Say do this for a group of (small) mersennine primes, record how long it takes to determine primality with your software/hardware setup, plot the points, then extrapolate how long it should take to compute the other known ones, see if they agree with known times, then suggest how long it would take for newer ones.

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That doesn't sound completely "not-fun" but alas I don't have an over abundance a time And while I did get a free Mathematica license from my school I haven't installed it yet because my computer is well.. sub-par. :) maybe this summer..

Wasn't it proven that there is no function that returns a prime for every integer? Aside the wow factor, is there an application for such discovery? :D

Whether it was proven that one cannot exist I'm not entirely sure, but as fas as application, there is a foundation offering a substantial sum of money to find one, 1 million dollars I think? :)

Borek
Mentor
Wasn't it proven that there is no function that returns a prime for every integer?
Unless I misunderstand what you mean, implementing function that will take n as a parameter and will return n-th prime, is trivial. Actually it was already done, many times. See for example this thread.

I can't guarantee these functions will work fast, nor can I guarantee your computer will have enough memory to run them for every n, but these are technical details.

Yeah, I think the problem is you can't possibly test all values for n. But if you are asking about the expression I posted in the op, that doesn't always work and simply describes the form of a specific type of prime number.

Borek
Mentor
ROFL, I am just working on a program, so when I saw "function" I thought in terms of a program function, not a mathematical one.

pbuk