
2^74207281-1 is Prime - ramshanker
http://www.mersenne.org/primes/?press=M74207281
======
jlewallen
Glad I actually read the article because I never would have known this:

"The official discovery date is the day a human took note of the result. This
is in keeping with tradition as M4253 is considered never to have been the
largest known prime number because Hurwitz in 1961 read his computer printout
backwards and saw M4423 was prime seconds before seeing that M4253 was also
prime."

~~~
banku_brougham
So if a mathematical fact is known by one person, "it is known."

~~~
1024core
That's how existence proofs work.

~~~
Eduard
I have seen aliens.

~~~
jfoutz
And I'm sure you'll have no more trouble proving the existence of aliens than
the fine mathematician who was the first to discover a fact.

So, let's see your proof.

------
coldpie
Big numbers are fun. Here's some stupid stuff I thought of while reading about
this.

The number is 2^74,207,281 - 1. Naively stored like a normal integer in
memory, that's roughly 75 MB of RAM just to store the number. Compare that to
your 64-bit time_t and imagine how long a duration of time could be
represented by such a large number. Stored on 3.5" floppy disks for some
reason, that's roughly 50 disks just to store the number.

Edit: Duh, bits, not bytes. I leave my mistake here for posterity.

~~~
cousin_it
Not unless you gzip it. Since the number is just the binary 1 repeated a bunch
of times, it should gzip pretty well.

~~~
biot
gzipped it's a 9043 byte file when the source is the number expressed exactly
as bits. gzipping that 9043 byte file yields a 129 byte file, which gzip can't
compress any further. Base64 of the file:

    
    
      H4sICBPHnlYCA2EuYmluegDt0LENQQEUQNHnK7xINF8rLIAVbKH88VU0ZlATK4gt1DQ6vURMoBID
      iN4CknOa29/BNrN72U+LZj2eL1fxPDU6EXE+3CbvUTEMAAAAAAAAAODfPfq9/DZfx6petw2B32Z5
      XZT31mYXH9vDv0RTIwAA

~~~
Houshalter
I see some redundancy there. The letter "A" appears many times in a row.

Also it could be compressed even further. All you need is a computer program
that prints "1", 74 million times.

~~~
elenorelange
It's already been concisely expressed as 2^74,207,281 - 1, so you don't need a
computer program that prints "1" 74 million times.

You just need a computer language that can interpret bignum math equations.

~~~
Houshalter
That would take ages to compute. Printing one is faster. And might produce the
smallest file size if it's done in assembly.

~~~
elenorelange
but the metric was the most concise way to store the data

------
JoeAltmaier
If you were abducted by aliens, and had memorized that number, you could
possibly trade it for wealth in their society. If they hadn't discovered it
yet!

~~~
yen223
If an alien were abducted by humans, and it memorized a much larger prime
number, what would we give in exchange for that piece of information?

~~~
kazazes
Apparently a $3,000 GIMPS research discovery award. What's the practical
utility of knowledge of a prime, especially considering that upon announcement
it's immediately disseminated?

~~~
contravariant
It's a very large prime number which has a very convenient representation,
which makes it useful. You could use it in a Mersenne Twister RNG, for
instance.

------
kylnew
I'm curious to understand the significance of finding the largest known prime
numbers. It seems that some people get pretty excited but I'm not sure if it's
just purely academic or there's some useful applications I'm unaware of. Can
anyone enlighten me?

~~~
ravanpao
I am curious too. I wonder if this has anything to do with the public key
encryption relying on prime numbers.

EFF has a bounty for large prime numbers:
[https://www.eff.org/awards/coop](https://www.eff.org/awards/coop)

 _" The EFF awards are about cooperation," said John Gilmore, EFF co-founder
and project leader for the awards. "Prime numbers are important in mathematics
and encryption, but the real message is that many other problems can be solved
by similar methods."

Finding these prime numbers will be no simple task, given today's
computational power. It has taken mathematicians years to uncover and confirm
new largest known primes. However, the computer industry produces millions of
new computers each year, which sit idle much of the time, running screen
savers or waiting for the user to do something. EFF is encouraging people to
pool their computing power over the Internet, to work together to share this
massive resource. In the process, EFF hopes to inspire experts to apply
collaborative computing to large problems, and thereby foster new technologies
and opportunities for everyone._

~~~
tinalumfoil
> However, the computer industry produces millions of new computers each year,
> which sit idle much of the time, running screen savers or waiting for the
> user to do something.

Considering how much more electricity a running computer uses versus an idle
one and how much faster consumer CPUs degrade under load this definitely seems
like a huge waste of resources for something that has little, if any, utility.

~~~
organsnyder
I used to run prime95 on all of my machines, back when power management
(especially on desktops) was virtually nonexistent. Obviously, with the power
management features we have now, an idle CPU (especially outside of a hosting
environment) is not considered wasteful.

------
agentgt
I had to find out what a Mersenne number was again (vaguely remember but
forgot) and I was shocked to find this problem in mathematics:

"It is not known whether or not there is an odd perfect number, but if there
is one it is big! This is probably the oldest unsolved problem in all of
mathematics." [1]

[1]:
[http://primes.utm.edu/mersenne/index.html](http://primes.utm.edu/mersenne/index.html)

------
haberman
> The primality proof took 31 days of non-stop computing on a PC with an Intel
> I7-4790 CPU.

How does this work? You just try to divide by every possible number and ensure
the result isn't an integer?

~~~
ajuc
You only need to divide by prime numbers up to ceil(sqrt(your_number)).

They don't know all prime numbers up to this number, but they can easily
exclude many numbers (for example even).

~~~
kale
That would still take an eternity.

They use special form primes (the big primes are all n+1 or n-1 forms, with n
being easy to factor).

If a Mersenne Number is prime, then the exponent must also be prime
(74,207,281 is also prime). That cuts down on a lot of processing time.

Also, all factors of a Mersenne Number, if it's composite, are of the form 2
_n_ k + 1, where the Mersenne number is written 2^n - 1. That also helps
factor it. I generated a huge list of prime number indices and did some tests
on them a couple of years ago. 25% of the Mersenne numbers have 2*n + 1 as a
factor. That's even less computation time to find a large prime.

------
pluteoid
Nice. Only the 49th Mersenne yet discovered. This also gives us a new perfect
number, 2^(74207281-1)(2^74207281-1).

------
gitpusher
Meanwhile, somewhere in Wisconsin, the number 274-207-2811 is being
mercilessly prank-called my mathematicians.

------
ramshanker
22,338,618 digits long, 1 More step towards EFF award for first 100 Million
digit prime number.

~~~
Yrlec
I wonder how much it would cost to generate that using EC2 spot instances.

~~~
daveguy
Amazon would like for you to try.

------
jacquesm
Wow, many years since I've contributed to the project with my little cluster,
nice to see them still going strong.

Incidentally, the project source code contains one of the fastest Fourier
transform implementations that I'm aware of.

~~~
nitrogen
Do you happen to know how it compares to kissfft or fftw?

~~~
jacquesm
Good question, no, no idea but of those two fftw would appear to have the edge
so if you're going to compare do it with that one. I do know the fftw guys
benchmarked an older version of it but I can't find the results (there used to
be a bunch of graphs showing the relative performance of all those ffts
listed).

It should be fairly easy to figure this out though, the benchmark code is
public and if fft sizes you are interested in are known you could make a very
precise comparison.

One warning about the mersenne fft code, it is not the most readable one on
account of all the optimizations, but if all you're interested in is a drop-in
library then it might suit your needs, especially if the size fft you're after
is one that has already been optimized to the hilt by the mersenne project
people. If you find you need to customize it then you're probably better off
with a more universal implementation, even at the expense of some cycles.

------
Consultant32452
I have a stupid question that will reveal how I know nothing about
cryptography. I understand that large primes are somehow important for some
types of cryptography. How are there not just databases of large primes that
make breaking the types of cryptography that rely on large primes trivial?

~~~
laughinghan
There are too many. Not just more than could fit in every database on Earth.
More than could fit on every database if every star in the sky had an Earth,
each with as many databases as there are on Earth.

This is nothing specific to cryptography or prime numbers. Fun fact: if you
shuffle a deck of cards correctly, it's almost certain that no one in history
has ever had a deck in the exact same order.
[https://www.math.hmc.edu/funfacts/ffiles/10002.4-6.shtml](https://www.math.hmc.edu/funfacts/ffiles/10002.4-6.shtml)

~~~
Consultant32452
I understand that there's a huge number of primes, but surely the primes being
used to generate keys or whatever process they're involved in are "known"
primes, right? It's not as if when I try to encrypt something my machine is
going to search for a new prime number that hopefully won't get guessed.

~~~
yoha
Long prime numbers are used for RSA keys. You only generate RSA keys when
creating a new identity (webserver certificate, client certificate, SSH key,
etc). Then, you _do_ search for "a new prime number that hopefully won't get
guessed". It can take from a few seconds to several minutes, depending on the
hardware and the desired key length.

The usual approach is to take a random odd number of desired length and then
have a probabilistic primality test (much faster than what would be needed for
a formal proof). Several attacks rely on bad entropy or bad pseudo random
number generators (PRNG) to guess what prime was selected.

------
hinkley
Are there any double Marsenne primes that we know of? Where 2^(2^n - 1) - 1 is
prime as well?

~~~
evandijk70
2^3-1 = 7 => prime 2^7-1 = 127 => prime 2^127 - 2 is also prime

------
legohead
is there anything special you can do with mersenne primes? if there's no
pattern to them, can't it just be seen as dumb luck?

~~~
syncsynchalt
They represent the largest known primes, as the special case 2^n-1 allows us
to verify their primeness using a method much more efficient than checking all
potential factors.

So to answer your question, if there's ever a need for ridiculously large
numbers that are known to be prime this is how we'll fill it.

Short of that need, it's still fun as a mathematical world record!

------
banku_brougham
I love this. So M49 is now the largest known prime, but what does it take to
find the next prime number? Or any larger prime?

~~~
dexterdog
A buttload of electricity

------
technotaoist
I downloaded and checked the zip file. Last digit is a 6. Downloaded it
multiple times to test. Can anyone else replicate this?

~~~
i_cannot_hack
If I have read it correctly, the last digit is a 1. This is the end of the
file I downloaded:

...010073391086436351

Edit: Confirmed by
[https://www.youtube.com/watch?v=q5ozBnrd5Zc](https://www.youtube.com/watch?v=q5ozBnrd5Zc)

~~~
technotaoist
I'll try a different tool to decompress the file maybe. a 6 would have made no
sense. What was your uncompressed file size? I ended up with 44.7MB

~~~
wrboyce

      % tail -1 M74207281/M74207281.txt
      010073391086436351
      % du M74207281/M74207281.txt
      44504	M74207281/M74207281.txt
      % md5 -q M74207281/M74207281.txt
      53294de131970c2cee8da6fe9a3135b3

