# 45th Mersenne prime discovered (possibly)

Distributed computing hasn't folded us a cure for cancer yet, but these projects in which PC users donate their spare processing power to solve scientific problems have unquestionably made major strides in uncovering ridiculously large prime numbers. Fans of bountiful digits may soon be rocked by the potential discovery of the …

#### I followed the link, but still don't know

While I can understand that a 10 million digit long prime number is a somewhat unique item, can someone elaborate on why it is worth $100,000? (well, aside from just the EFF putting that bounty on it) I know primes are used in encryption, but 10 million digits seems, to me, too long to be useful...

#### I think 9 < 10 is still true

"the 44th Mersenne prime is 232,582,657-1, which works out to be 9,808,358 decimal digits long.... The 45th Mersenne prime may qualify for a $100,000 prize ... to anyone who discovers a prime number with at least 10 million digits."

Um, last time I looked, 9 is less than 10, and by standard mathematical rules, 9 *million* (and change) is still less than 10 *million*.

So just how could they have a valid claim on the prize? No - the assertion that 0.99999... == 1 doesn't wash in this instance.

#### Primes?

Erm, sorry to be utterly anal and pedantic, but I'm sure that 15 and 63 aren't prime, being 3*5 and 7*9 (or 7*3*3) repectively?

Mine's the one with the pockect calculator in (guess where?) the pocket.

--Martin

#### @ Del Merrit I think 45>44

The 44th Mersenne prime was 9808358. They think they have found the 45th. And *that* number could very well be 10M digits.

#### Re: I think 9 < 10 is still true

So, by your cunning reasoning, you demonstrate that the 44th Mersenne prime, which has less than 10 million digits, doesn't qualify for the prize. Well done. The prizegivers obviously thought so to, otherwise they'd have awarded the prize already.

The 45th prime is going to be bigger than the 44th. The 44th was 650,000 digits longer than the 43rd. This does vaguely suggest that the 45th will have over 10 million digits.

Do pay attention.

#### Primes?

Oopsie - should have read the article a bit more closely first; the it clearly differentiates between "Mersene numbers" and "Mersene primes".

Shuffles away in embarrasement...

--Martin

#### @Del Merritt

The assertion seems to be that, as the 44th Mersenne prime is just under 10 million digits long, the 45th Mersenne prime may well be over 10 million digits.

#### "He estimates completion on September 12 and September 16"

Shouldn't that be September 11 and September 15??

I'll go now...

#### @Del

Doubt I'm the first one to say this but ...

It was the 44th Mersenne prime that is 9,808,358 digits long, and the 45th that might be more than 10 million. That's 45, not 44.

#### What would be impressive

would be someone discovering this number using mental arithmetic.

Perhaps Paris should be tasked with finding the 46th.

#### Pants

That's what it is: pants. This isn't a big accomplishment - anyone with enough CPU's could have done it. The reason that GIMPS did it is because they are a bunch of beggars whose project couldn't get enough funding to solder together a bunch of PS3's and come up with the number on their own. Pants.

#### Nice to see...

...such clever minds take time out from other pursuits like finding cures for cancer, sustainable food production, and how to increase my bandwidth so that I can download porn faster than mankind can create porn sites to host it all on?

OK, joking aside, really, what's the point?

Paris - porn, she knows what I'm talking about.

#### @@Del Merrit,

Del Merrit is clearly joking ... he even gives you the hint by asserting that zero point nine recurring is not equal to one. Only jokers and fuckwits say that.

#### Send the money now!

I have calculated the largest prime, also the largest Mersenne prime: infinity. You may send me my check now.

#### More pedantry please

What may have been found is the 45th *known* Mersenne prime. Saying just "the 45th Mersenne prime" means the 45th smallest, and it's at least somewhat plausible that there are Mersenne primes that, through accident or malice, GIMPS missed.

So for example, M(32582657) is the 44th Mersenne prime to be discovered, and the 44th smallest of the known Mersenne primes, but that doesn't necessarily mean it is, as the Reg claims, the 44th Mersenne prime. M(127) was proved to be prime before M(61) was, but that doesn't make M(127) the 9th Mersenne prime and M(61) the 10th.

It's as certain that those found BY gimps really are prime, btw, as we can be of any computer proof, since the verification process doesn't rely on thousands of people on the internet to all tell the truth. So these pair aren't any *less* than the 44th and (assuming verified) 45th Mersenne primes. But they might be more.

#### M45 may be less than M44

The process of searching for the possible new Mersenne prime is non-sequential. There's always a chance that the 45th *found* number would be in fact less than the previous one(s). That has only happened before with very "small" Mersenne numbers (centuries ago), and not yet in modern times. You can check the Wikipedia http://en.wikipedia.org/wiki/Mersenne_prime -- see numbers in the first dozen.

#### Ohh if you real want to spice up the conversation with a mathematician

Just claim that 9.9999 recurring is not equal to 10, because the 9 is not 10, so there will be a very very very small difference. Then ask them if they also hold that 1.11111111 recurring in base 2 is equivalent to 10.

Oh, watch the mathematical sparks fly :)

#### Confirmation Date

Mersenne prime + Large Hadron Collider = 10th September is end of the world.

#### And now, for something completely.... errr, pointless?

Fine, OK, so maybe not pointless.

But as stated earlier, I would be REALLY curious to know why millions of computer-hours were invested in THIS instead of something more obviously beneficial to mankind...

#### Re: Send the money now!

I think you'll find that infinity is divisible by an infinite number of numbers.

#### @Steve VanSlyck

>> I have calculated the largest prime, also the largest Mersenne prime: infinity...

I counter that, coz I've just come up with a bigger number, which is your number (infinity) plus 2......

So, you can counter sign the cheque and send it on to me....

PH.....coz she's worth every penny......

#### That's perfect of course

As Euclid proved llong ago, if M(n) is a Mersenne prime, then the larget number P(n) given by (4^n - 2^n)/2 is a perfect number. Thus we have:

M(2) = 3, P(2) = (16 - 4) / 2 = 6

M(3) = 7, P(3) = (64 - 8) / 2 = 28

M(5) = 31, P(5) = (1024 - 32) / 2 = 496

M(7) = 127, P(7) = (16384 - 128) / 2 = 8128

These numbers are described as perfect because in each case the sum of factors is equal to the number itself, e.g. 28 = 1 + 2 + 4 + 7 + 14.

The new discovery of the 45th Mersenne prime therefore means ipso facto the discovery of the 45th known perfect number.

In later days Euler showed that Euclid's schema is the only schema for even perfect numbers. (Odd perfect numbers are an open question - it's known that there are none of 300 digits or less.)

(For the record, numbers where the sum of factors falls short of the number - the majority - are called "deficient", numbers where the sum of factors exceeds the number are known as "abundant". The first abundant number is 12. Odd abundant numbers are possible, the smallest being 945.)

#### @AC

>Just claim that 9.9999 recurring is not equal to 10, because the 9 is not 10

>so there will be a very very very small difference.

No difference whatsoever.

>Then ask them if they also hold that 1.11111111 recurring in base 2

>is equivalent to 10.

No, it's equivalent to 2. Or 2r10

There you go, no need for any mathematical sparks whatsoever. I am perfectly prepared to explain why zero point nine recurring is equal to 1 to any person who is curious. However, if they keep insisting I am wrong the conversation is over. It's pretty much the same with creationists, although to be fair they have a bigger chance of being right.

#### @Sleeping Dragon,Andre Carneiro

The point is knowlege for its own sake, our species wouldn't be where it is today if all scientific endeavor was based solely on short termism.

#### optimus prime

"...according to GIMPS protocol, the potential 45th Mersenne prime will not be officially disclosed until it has been verified as a prime... estimated completion on September 12 and September 16..."

aw! - c'mon reg. how can you run this story now, knowing you'll leave us all at the end of our tethers with suspense til the middle of september. i know i shan't get a wink of sleep until i hear the outcome of the verification process!

#### Better process for finding prime numbers

If you want to find large prime numbers, you don't need a computer. Just a paruresis sufferer, a large quantity of supermarket own-brand diet lemonade and a toilet where the occupant can hear every sound outside.

#### @AC

"Mersenne prime + Large Hadron Collider = 10th September is end of the world."

Well thank fuck for that. This will save me a load of hassle.

Anyone who thinks infinity is a prime number is kidding themselves.

Infinity/2 = Infinity

Infinity/3 = Infinity

Infinity/4 = Infinity

Shall I go on?

Or shall I get my coat?

#### You can see why finding these numbers have such a cult following.

No, I really can't.

Paris because she'd rather suck dick than waste time on Mersenne Primes.

#### We know that testing for primality is in P

...so how many operations does it take to test this extra-large number? Or will testing be done probabilistically?

#### @DAM

LOL Probabilistically - in my sleep-addled state, I read that as Probaballistically, which would be something like the chance of getting shot...

#### @ Peter Timon

Oh really, does it solve World Peace? Can we generate 0-carbon electricity from this discovery?

No? Then why bother?

What's the point of discovering n th precision of a number or equation if it does nothing to preserve humanity or the world around us?

You might as well collect stamps and count then number of Penny Blacks you have.

Paris, because even she isn't such a waste of space.

#### to Andre Carneiro

"...I would be REALLY curious to know why millions of computer-hours were invested in THIS instead of something more obviously beneficial to mankind..."

Well why not? Why did you invest a minute of your time posting a comment when you could have done something more beneficial. If a load of people want to crunch prime numbers for fun there's nothing wrong with that. Blimey, next you'll be outlawing music.

#### Easiest way to prove 0.99999.. equal to 1

For all the non-believers (tho most I'm hoping to be joking), the easiest way to prove 0.99999 recurring equals 1 is to consider how to represent it as a fraction.

1/9 = 0.1111111111 recurring

0.99999999 recurring = 9 * 0.11111111111 recurring.

So you have 9 x 1/9 = 1.

Or, more formally:

x = 0.9999...

10x = 9.9999...

10x - x = 9.9999... - 0.9999...

9x = 9

x = 1

#### Actually it's easier

To know if a number N is prime, you only have to test numbers less than or equal to sqrt(N). And if you know all the primes up to that, it's a LOT easier, as you only need to test if the number is divisible by the primes less than or equal to sqrt(N). Granted, if you don't know the primes up to sqrt(N), it's a lot harder.

It's very simple if you think about it.

Finding primes is one of those problems that rapidly expands into however much computing power you can throw at it, If you know all primes up to M, you can start working on everything from M+1 to M^2, which gets large fast.

#### @Sleeping Dragon

How do you know it won't have some kind of obscure but crucial bearing on your (to you) more important projects?

i don't suppose Einstein knew that his work would affect just about every aspect of our civilisation.

Knowlege for its own sake doesn't need justification from anyone.

#### You can't divide infinity by 2

>Anyone who thinks infinity is a prime number is kidding themselves.

>Infinity/2 = Infinity

>I think you'll find that infinity is divisible by an infinite number of numbers.

When talking about prime numbers, the work "divide" refers to the operation of integer division. This operation is definted on .... integers. "Infinity" isn't an integer. It isn't a rational number, it isn't an irrational number, it's not a real number, and it's not a complex number. So all three of the statements quotes above are basically nonsensical.

There is of course a branch of mathematics called transfinite mathematics, which works with the different types of infinity - but anyone who works in that field will quite happily tell you that infinity is not a number. If you're engaged in a discussion of numbers, starting to talk about infinity will almost always result in confusion and error, unless it's clearly understood that any statement about infinity can be rewritten without using the term (e.g. "there are an infinite number of primes" -> "there is no integer which represents the total number of primes. When you think you've got them all, it's quite easy to prove there is at least one more".

#### It's not that easy!

This method doesn't work for numbers of the order of magnitude of, e.g. 9,808,358 digits denary.

Where are you going to store all the primes less than the SQRT of this number?

#### @Chris

Dude, chill out! I'm not trying to outlaw anything!

(Although RnB should be) ;)

I was really just curious to find out if there was any particular application to this piece of knowledge that wasn't immediately obvious, that's all.

And it's very kind to compare the value of one minute of my time to all those computer hours. I'm flattered. :P

#### infinities of all shapes and sizes

A bit of a curiosity is infinity. There are infinite natural numbers; 1, 2, 3, 4, ... etc.

There are infinite rational ( 1/4, 0.25) and irrational numbers (pi, sqrt 2)

There are an infinite real numbers, values between 1 and 2, 2 and 3 etc.

A set of all the real numbers is infinitely larger than the set of all rational numbers which is in turn infinitely larger than the set of all the natural numbers. It's elephants all the way down.

Infinities come in different sizes.

#### Prime testing

This method - testing all the primes < SQRT(x) - is still (subject to a few tweaks) the most effective method for *factorising* a given number, at least until someone contructs a working quantum computer. There are much swifter techniques for testing if a number is prime * (particularly numbers of the form 2^p-1), so we are able to say "this number is not prime" without having any idea what its factors are.

* some of the more efficient techniques are known to generate false negatives (very occasionally). So sometimes when you generate large primes (e.g. for use in RSA cryptography) one of the numbers may be only a pseudo-prime, in which case the encryption will fail. This happens rarely enough that the gains in speed are considered worthwhile compensation.

#### Two corrections

The previous record-holding Mersenne Prime wasn't 232,582,657 - 1, which would be an even number; it was (2 to the power of 32,582,687) minus one.

Also, while Cantor's diagonal proof shows that there are more real numbers than there are natural numbers, the rational numbers and the natural numbers (or the integers) have the same cardinality.

#### @Peyton

It's not "somewhat unique." It is actually unique. A thing is unique or it is not, period. There aren't any shades of grey.

#### @Paul Buxton and @AC

The end of the world is not in September. It is in December. 12/12/2012, to be precise. As to everybody else, .999 &c. clearly is equal to 1. I have Excel open right now and it says so right there.