
A New Largest-Known Prime Number - tshannon
https://www.npr.org/2018/12/21/679207604/the-world-has-a-new-largest-known-prime-number
======
userbinator
_We would write the number out for you, but it would fill up thousands of
pages, give or take, and look like this gigantic zip file._

It's ironic to see that 11MB file being called "gigantic" in this day and age;
many web pages are now unfortunately much larger, and the page of the article
itself transfers at least 1MB of data.

~~~
carlob
Then it's probably in decimal form. The binary representation should compress
much better :)

~~~
dvh
Any number N in base (N) is written as 1

~~~
carlob
no it's 10

~~~
SomeHacker44
Typical programmer. Off by zero as always.

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delhanty
Read this and wondered, what estimates are there on the density of the
Mersenne primes as n grows?

Google told me that, surprisingly, it's not even known that they are infinite!

I found this nice page on the Wagstaff Mersenne Conjecture:

[https://primes.utm.edu/mersenne/heuristic.html](https://primes.utm.edu/mersenne/heuristic.html)

~~~
ryacko
Possibly the only civilian use of quantum computing, an interesting decade is
ahead.

~~~
scythe
Primality testing is actually much easier than integer factorization thanks to
Fermat's "little" theorem. Methods based on this theorem, including the Lucas-
Lehmer test for Mersenne primes, can identify prime numbers on classical
computers in polynomial time. However, factorization is not possible with this
method.

~~~
chris_st
That's okay, hardly anyone needs to factor prime numbers :-)

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fluxty
I used to be into this stuff in college--we were obsessed with the idea of
generating very large or else very large numbers of odd numbers, such as weird
numbers or untouchable numbers. I was into HPC and writing programs that run
on clusters, and this gave me some good practice.

We eventually realized that no one really cared about the numbers we felt
confident we could find--now Mersenne primes, though, people really care about
those, but they're extraordinarily difficult to find. Always a cool day to see
a new one.

~~~
AlexCoventry
It's not clear to me that anyone really cares about Mersenne primes, either,
which is not to detract from this cool result in any way.

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heyjudy
Mersenne primes in binary are all 1's with length _n._

PS: I remember scheduling Prime95 to run on lab computers at work (a nuclear
engineering shop) after-hours, c. 1998.

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NKosmatos
The official press release from GIMPS:
[https://www.mersenne.org/primes/?press=M82589933](https://www.mersenne.org/primes/?press=M82589933)

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eximius
Anyone know what primality test they use? Miller-Rabin for testing then
something deterministic for verification?

~~~
ehsankia
They use Lucas-Lehmer's[0] which is specifically for Mersenne primes[1]

[https://www.mersenne.org/various/math.php#lucas-
lehmer](https://www.mersenne.org/various/math.php#lucas-lehmer)

[https://en.wikipedia.org/wiki/Lucas%E2%80%93Lehmer_primality...](https://en.wikipedia.org/wiki/Lucas%E2%80%93Lehmer_primality_test)

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man-and-laptop
It would be nice if this were used as the PoW problem for a blockchain. Since
the Mersenne Primes are rare, the PoW problem shouldn't be to find new
Mersenne Primes directly, but something that could make indirect progress on
that. Also the blockchain should solve an actual problem, such as torrent
tracking or DNS.

It makes me think that there should be a list of viable uses of blockchain,
and another list of interesting problems that could be moulded into PoWs, and
they should be paired up into blockchains.

~~~
topmonk
One problem is that a black hat could solve a lot of primes in secret, and
then use those for a double spend attack.

Specifically,

a) Spend some coins to buy some other cryptocurrency

b) use all their secret primes creating a bunch of new blocks on a branch
where they haven't spent their money. Since this is the longest branch now,
all the other clients accept they haven't really spent it.

c) buy the other cryptocurrency again on this new branch.

~~~
bouncycastle
Actually, the problem is that there is no way to control the difficulty.
Finding new primes would be harder and harder, but you want the difficulty to
adjust up or down to target a block time. Not sure how primecoin approached
that problem?

Update: found some clues here,
[https://www.reddit.com/r/primecoin/comments/1i71ma/strange_b...](https://www.reddit.com/r/primecoin/comments/1i71ma/strange_behaviour_of_difficulty_adjustments/)

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burtonator
.

~~~
thestoicattack
Two is prime.

~~~
zahreeley
According to German science magazine 2 is not prime

~~~
heyjudy
That's not a primary source and wrong. A prime is most simply: it has no
factors other than itself and 1. "Not being even" is a corollary, and would be
redundant if included in the definition of primality.

~~~
schoen
Another way to see this is to observe that the Fundamental Theorem of
Arithmetic would be wrong (or more difficult to state succinctly) if 2 were
not considered prime or if 1 were considered prime.

For example, if 2 were not prime, it would be _impossible_ to represent the
number 256 (or the number 123456) as a product of primes.

~~~
throwawaymath
This is an excellent rebuttal, nice.

~~~
lurchedsawyer
It's not, though. It's one group of mathematicians appropriating prime numbers
and jack-booting them into their favourite theorem. There are valid reasons to
treat 1 as a prime number, just as there are for defining 0^0 as 1 in some
cases. Also for a fun pastime try asking an arithmetician to apply the
fundamental theorem to 1 and watch them squirm.

~~~
schoen
Presumably the arithmetician will answer that its factorization is the empty
set. That's what sympy says, for example.

    
    
      >>> sympy.factorint(6)
      {2: 1, 3: 1}
      >>> sympy.factorint(1)
      {}
    

This definition is also sensible because it also preserves uniqueness in both
directions. The empty set is the only prime factorization of 1, and 1 is the
only natural number whose prime factorization is the empty set. (Wikipedia has
a footnote that "Using the empty product rule one need not exclude the number
1 [from the fundamental theorem of arithmetic], and the theorem can be stated
as: every positive integer has unique prime factorization.")

The fundamental theorem doesn't have to state that the prime factorization is
_nonempty_.

It's true that some mathematicians have defined 1 as a prime number and
there's nothing logically inconsistent about doing so, but it makes most
theorems and formulas in number theory more complex and so this definition has
fallen out of favor.

Edit: I think the Wikipedia article on the empty product gives some quite nice
examples of the benefits of a closely related concept.
[https://en.wikipedia.org/wiki/Empty_product](https://en.wikipedia.org/wiki/Empty_product)

~~~
impendia
> Presumably the arithmetician will answer that its factorization is the empty
> set.

Arithmetician (Ph.D. in number theory) here. This answer is completely
correct.

As another reason why the factorization of 1 should be the empty set: suppose
you have two positive integers m and n. Write S(m) and S(n) for their sets of
prime divisors, counted with multiplicity.

Then S(mn) is the union of S(m) and S(n). We need S(1) to be the empty set to
make this rule consistent.

Analogous to how log(1) is equal to 0.

~~~
Someone
It also is analogous to the fact that the sum of zero integers equals zero (a
claim that, I guess, more laymen will find reasonable)

The analogy is a good one, as zero is the additive identity
([https://en.wikipedia.org/wiki/Additive_identity](https://en.wikipedia.org/wiki/Additive_identity))
of the integers, and one is the multiplicative one
([https://en.m.wikipedia.org/wiki/1#Mathematics](https://en.m.wikipedia.org/wiki/1#Mathematics))

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revskill
Number Theory is beautiful, but impractical for daily usage or programming.
Who cares ?

~~~
dang
"Please don't post shallow dismissals, especially of other people's work. A
good critical comment teaches us something."

[https://news.ycombinator.com/newsguidelines.html](https://news.ycombinator.com/newsguidelines.html)

