Largest Twin Prime Yet Discovered

Chris Chiasson writes "The Twin Internet Prime Search and PrimeGrid have recently discovered the largest known twin prime. A twin prime is a pair of prime numbers separated by the integer two. The pair discovered on January 15th was 2003663613 * 2^195,000 ± 1. The two primes are 58,711 digits long. The discoverer was Eric Vautier, from France."

Comments

Are you kidding?

[greg_barton]

A twin prime is a pair of prime numbers separated by the integer two.

Are you kidding? Those are easy to find! Try getting two primes separated by the integer three...

Re:Are you kidding?

[EmagGeek]

You mean like this?

137

The primes are 1 and 7, separated by the integer 3...

Re:Are you kidding?

[fredmosby]

How about 2 and 5.

Re:Are you kidding?

[Peter Cooper]

Sorry to take a dump on a cute joke with pedantry, but 1 isn't a prime.

Re:Are you kidding?

[SeanMon]

1 isn't a prime.

Re:Are you kidding?

[proverbialcow]

Very good. Now try finding two primes whose difference is 7.

And when you're done with that, find two perfect cubes whose difference is also a perfect cube. I did this once, but there wasn't enough room in the margin to write the answer.

Re:Are you kidding?

[Anonymous Coward]

Very funny, Fermat's last theorem was already proven.

A^n + B^n = C^n
A^n + B^n - C^n = 0
A^n - C^n = - B^n
C^n - A^n = B^n

Since 3 isn't 2, then I assume that what you said isn't possible.

Re:Are you kidding?

[cperciva]

Now try finding two primes whose difference is 7.

How about 5 and (-2)?

Re:Are you kidding?

[Anonymous Coward]

I thought a twin prime was ... bada-bing! - Anna Nicole Smith.

Re:Are you kidding?

[greg_barton]

Gods, people. Is there no humor left in the world of mathematics? Well, at least not slashdot math geeks.

OK, here's the joke. Yeah, 2 and 5 are primes separated by 3. There aren't any others because all primes other than 2 are odd, and adding 3 to an odd number results in a composite number, which can't be prime.

So you'll be searching for such primes forever. Get it?

Jeez. Apparently the joke is ya'll searching forever for your sense of humor...

Re:Are you kidding?

[Manatra]

Actually, it depends on the person you talk to. Some people consider the number 1 a prime, some don't.

(for the record I don't treat 1 as a prime number)

Re:Are you kidding?

[Gobiner]

And when you're done with that, find two perfect cubes whose difference is also a perfect cube. I did this once, but there wasn't enough room in the margin to write the answer.

My answer is 3 and 7.

Who are you to tell me what numbers are perfect and what aren't? I shall decide for myself, in the way of my fathers.

Re:Are you kidding?

[DahGhostfacedFiddlah]

Nice try.

Re:Are you kidding?

[Chacham]

1 isn't a prime

Yes, it is.

Re:Are you kidding?

[Propaganda13]

What's the earliest definition of prime numbers?

I've read several definitions over the years. Some read as if 1 could be prime (divisible only by 1 and itself), some specifically exclude 1 as a case, and some definitions like Wikipedia (if I don't go edit it ;)) point out two distinct factors thereby excluding 1.

Re:Are you kidding?

[Anonymous Coward]

Mathematicians have been known to alter the primality of 1 based on convenience. Generally it doesn't matter very much whether you consider it prime or not.

Re:Are you kidding?

[Secret Rabbit]

To join this little debate (replying to you as I don't want to reply to two different people with the same post):

Actually, if one considers 1 a prime problems end up happening e.g. inconsistencies with algebraic number theory (prime ideals) and elementary number theory. Basically, if you pop in 1, elementary number theory is fine (at least up to where I've studied it doesn't really matter aside from making some proofs more difficult than necessary). But, then some further developments like algebraic number theory start having problems, like the before mentioned inconsistency in the definition of a prime.

Leaving 1 out as a prime makes the elementary number theoretic definition consistent with the algebraic number theoretic definition. Just thought I'd point that out as math is all about detail and consistency. And not having a consistent definition of a prime is a rather large f**k up as we all know how important primes are.

So, although 1 has been considered a prime in the past, it does seem (keep in mind, I've looked through several libraries) that 1 has been dropped as a prime. Modern mathematics seems to have taken care of this discussion.

Re:Are you kidding?

[cperciva]

Nice try.

Somehow I'm not surprised to find that materials written for consumption by grade school students (and teachers) get this wrong. A prime element of an integral domain is a non-zero non-unit p such that if p divides ab, p divides either a or b (or both). The integers are an integral domain, and (-5) is a prime.

Re:Are you kidding?

[saforrest]

Nice try.

Yeah, it was a nice try. A nice, successful try. Until I read the post, I was going to suggest 2 and -5, which would've worked too.

Generally speaking in algebra, any unit multiple of a prime is considered a prime. Because -1 is the only unit other than 1, the negative numbers aren't usually counted, but there's no good reason not to apply the more general definition here.

Re:Are you kidding?

[Anonymous Coward]

(A = 0 and B = C) or (B = 0 and A = C)

Re:Are you kidding?

[Garridan]

Same goes for -1. In fact, John Conway, among others, considers -1 to be prime. Good math books contain rigorous definitions of any terms used. If I want to use the word "prime" to denote "any natural number that is 5 greater than another natural number", that's my business. Others may not like the terminology, but if the math is good, the result is good.

Math & written language must coexist, but at the same time, the line between them must not be blurred.

Re:Are you kidding?

[xouumalperxe]

2 and 5
Do I get a cookie?

Re:Are you kidding?

[Garridan]

The definition of prime numbers varies quite a bit, depending on the application.

Similarly, many don't even agree what log(x) means!
For high school math, log(x) is log base 10.
In undergraduate math, and statistics, log(x) is the natural log.
Later on in math, log(x) is of the most convenient base for the application, unless this is ambiguous or non-obvious. In computer science, log(x) is frequently base 2, but nobody really cares 'cause change of base is just multiplication by a constant.

Re:Are you kidding?

[rbarreira]

Calm down. The guy probably got the joke, and was just pointing out that there is *one* such pair of primes... Nothing bad in my opinion.

Re:Are you kidding?

[Criffer]

And when you're done with that, find a difference of squares that equals RSA-2048, and I'll split the $200k with you.

Re:Are you kidding?

[stupid_is]

Interesting - in my maths degree and at school (in the UK), we were taught that log(x) was base 10, and ln(x) was the natural log. Other ways of writing it would be to include the base as a subscript to the log(), which made it more obvious when doing those tedious exercises to convert the base.

Re:Are you kidding?

[eldepeche]

LOL

Re:Are you kidding?

[Anonymous Coward]

I'm no mathmatician but aren't 2 and 5 prime numbers seperated by an integer of 3? Or did I miss something here?

Re:Are you kidding?

[eldepeche]

Actually, everyone agrees what it means. It's just that different people have use for different bases, so if an analyst is talking to an analyst, they both know which log they're talking about.

And I'm pretty sure the only reason anyone cares about the base 10 log is because sliderules were invented.

Re:Are you kidding?

[Paradise Pete]

Very good. Now try finding two primes whose difference is 7.

Well I would say 7 and 14, but then nobody writes jokes in base 13.

Re:Are you kidding?

[DemoLiter3]

Theorem : there are infinitely many odd numbers n, for which at least one pair of primes ( x , x+n ) exists.

Re:Are you kidding?

[MindStalker]

Well they certainly don't have to be distinct, because then 25 would be prime. You mean two numbers that are distinct from the original.

Re:Are you kidding?

[Record Keeper]

(for the record I don't treat 1 as a prime number)

Noted.

Re:Are you kidding?

[fatphil]

"Now try finding two primes whose difference is 7."

4+w and 11+w in the Eisenstein Integers. (so w is the primitive cube root of unity)

Re:Are you kidding?

[fatphil]

Bollocks.

2+w and 9+w.

Arse.

Re:Are you kidding?

[Anonymous Coward]

2 & 5
wow

Re:Are you kidding?

[Anonymous Coward]

I think that's 2 & 5 and rather easy at that...

Re:Are you kidding?

[Anonymous Coward]

How about 2 and 5?

Re:Are you kidding?

[mstahl]

2 and 5?

Re:Are you kidding?

[collectivescott]

"Interesting - in my maths degree and at school (in the UK), we were taught that log(x) was base 10, and ln(x) was the natural log. Other ways of writing it would be to include the base as a subscript to the log(), which made it more obvious when doing those tedious exercises to convert the base."

That's exactly how I learned it here in the US as well. The terminology remained the same through college level math classes as well as computer science.

Re:Are you kidding?

[sokoban]

] 1 isn't a prime. I guess next you're going to tell me that Optimus isn't a prime.

Re:Are you kidding?

[strikethree]

A twin prime is a pair of prime numbers separated by the integer two.

Are you kidding? Those are easy to find! Try getting two primes separated by the integer three... How about 2 and 5? :)

strike

Re:Are you kidding?

[jhantin]

A further shorthand notation I've encountered is lg(x) for base 2.

Re:Are you kidding?

[FunkyELF]

Wow, didn't realize why your's was rated Funny, but then I noticed your name. Very nice. If I had mod points I would rate you funny as well...not that it would mean much as this was obviously created for a one time use.

Re:Are you kidding?

[Anonymous Coward]

> And when you're done with that, find two perfect cubes whose difference is also a perfect cube. I did this once, but there wasn't enough room in the margin to write the answer.

How about 1 and 0? Plenty of room left in my margin...

Re:Are you kidding?

[DahGhostfacedFiddlah]

Might want to change this Wikipedia article then.

It seems to indicate that the general usage "prime number" implies Natural numbers, while different branches of math have an expanded definition.

Of course, the absolute, no-nonsense, definitive source (Merriam-Webster) says +/-, so who am I to argue?

Personally, I'm drawn to the mathforum.com link from my first post. Allowing negative primes would break some long-standing assumptions about primes, so don't allow them.

Re:Are you kidding?

[Secret Rabbit]

"""
Others may not like the terminology, but if the math is good, the result is good.
"""

That isn't exactly true. Mathematics, if it is to co-exist with other maths, it must be consistent. By altering definitions, there may be unintended consequences (if not only confusion).

An example of problems with redefining things, would be the first attempts at proving Fermat's last theorem. Basically, people were working in a system that they thought was fine, but in actuality they assumed that they had unique prime factorization which they actually didn't. This caused the proof to be erroneous.

It is currently established that a prime number is a number that is greater than 1 and has divisors that are only 1 and itself. You start messing with that and something else might go away, something that you need or something someone else needs that is attempting to use your work.

You just can't screw with established definitions. Now if you wanted to call your definition wozzle-wuzzle, no one's going to care. But, "Prime Number" has a clear established definition and no-one is going to take you seriously (never mind getting published) if you ignore that.

Re:Are you kidding?

[hepwori]

No, 25 has three distinct factors: 1, 5 and 25.

Re:Are you kidding?

[Garridan]

If base 10 wasn't useful, it wouldn't have been on the slide rule, either. It's useful for people who count in base 10. Quick way to count digits. If you give me a log in base 2, I've gotta divide by log_2(10) to see if we're talking about thousands, millions, billions.

Re:Are you kidding?

[EmagGeek]

Okay FINE... assuming 1 is not prime, which some argue it is...

237

Happy now? :)

Good example of a /. story.

[Ninjaesque One]

Succinct, on a subject undeniably nerdy, and mostly devoid of spelling mistakes. Also, not 'edited' by Zonk.

Re:Good example of a /. story.

[MagicM]

Wouldn't that make it a bad example of a /. story?

*rimshot*

Re:Good example of a /. story.

[gstoddart]

Wouldn't that make it a bad example of a /. story?

Nope, it's an uncommon example of a /. story which we wish was more common. :-P

Don't seem too excited

[presidentbeef]

The website announcement doesn't seem all that excited about the discovery.

Odd?

Re:Don't seem too excited

[odasnac]

generally, yeah. most prime numbers are odd.


...

i'm so sorry.

Re:Don't seem too excited

[cperciva]

most prime numbers are odd.

Only on slashdot would the parent get moderated as "informative"...

Re:Don't seem too excited

[cgibbard]

Finding twin primes like this is mostly just an elaborate computational game which doesn't really tell much about the mathematical structure of twin primes. It doesn't help at all with knowing whether there are infinitely many or not, for example. The same goes for other searches for large primes.

Also, if you're asking about real-world practical considerations, the primes used in practical work by comparison are tiny. Using such large primes for things like cryptography would be stupid for a number of reasons, not the least of which being that there are only so many known such primes out there, the size of your key would give it away. Personally, I don't know of any practical use for twin-primes or Mersenne primes, or any of the other classes of large primes being searched for.

It's really more just for fun, like computing digits of pi. However, devising new ways to access large twin primes, for instance, results in improvements of our knowledge of them. It's those new theorems and algorithms which people might get excited about. Running a computer for hours or days or months to actually find the things is less interesting. ;)

Re:Don't seem too excited

[Danny Rathjens]

most prime numbers are odd.

Only on slashdot would the parent get moderated as "informative"...

Do you know why 2 is odd?

It's the only even prime number. :)

Re:Don't seem too excited

[Anonymous Coward]

Only on slashdot would the parent get moderated.

There, fixed it for you.

Re:Don't seem too excited

[presidentbeef]

Hahahaha! :D
Thanks.

Re:Don't seem too excited

[Grey Ninja]

Thanks. I have milk all over my face now from trying not to spray it on the screen. I'm way too tired to be reading jokes this terrible. =P

Re:Don't seem too excited

[oliverthered]

I think Mersenne primes are good for psudorandom number generators.

How is this meaningful?

[JimMcc]

Seriously. I'm not a math major, etc. But I'm curious, is this of value? Other than of course as a curiosity.

Re:How is this meaningful?

[hamburger lady]

these numbers can totally come in useful in finding a cure for cancer.

Re:How is this meaningful?

[JimMcc]

Rrrrriiiiiiiigggggggggggggggghhhhhhhhhhhhhhhhhtttt tttttttttttttttttttt!

Re:How is this meaningful?

[0rionx]

This article is a pretty good summary of the reasons behind the search for large primes. Although finding a new large prime doesn't necessarily have any specific, short term "benefits", it serves to deepen our understanding of mathematics. As extremely large primes are of importance in cryptography, the ability to find and work with large primes has a great deal relevancy in IT, as well. The more we discover large primes the more we learn about ways to factor them quickly and efficiently.

Re:How is this meaningful?

[TravisW]

It depends on what you mean by "of value."

At any rate, any particular pair of twin primes is unlikely* to be especially "significant." However, an important open problem in math is, "Do there exist infinitely many twin primes?" Experts think it's likely enough that the answer is yes that they've named that supposition "Twin Prime Conjecture," which indicates that those experts consider it definitely less than a theorem but much more than a wild guess.

That the problem is so simply stated but remains unsolved is a testament to its difficulty (cf. Fermat's Last Theorem a.k.a. Wiles' Theorem). Hardy and Wright wrote to this effect: "The evidence, when examined in detail, appears to justify this conjecture, but the proof or disproof of conjectures of this type is at present beyond the resources of mathematics."

*If the conjecture is false, that is, if there are only finitely many twin primes, certainly the largest pair is important.

Incidentally, the "Pentium bug" was discovered when someone computed the reciprocals of two large (twin) primes and noticed an error after about 10 decimal paces.

Twin Prime (Wikipedia)

Re:How is this meaningful?

[rifftide]

They're doing it all for their kids.

Specifically: "My dad's useless numbers are bigger than your dad's useless numbers."

Re:How is this meaningful?

[Anonymous Coward]

It's value is about equivalent to the time you spend here in lamentations. At least these boys are doing something, besides pissing and moaning.

Re:How is this meaningful?

[QuantumG]

scoff all you want. You wouldn't believe the kinds of math that have been applied to gnome sequencing.. stuff that was discovered in completely different domains. That's the beauty of math.

Re:How is this meaningful?

[vga_init]

Well, it is generally believed that prime numbers are infinite... that is, we can count them and never run out. All this is an effort to see if we actually do run out. ;)

Re:How is this meaningful?

[jallen02]

I am VERY concerned about this gnome sequencing you speak of? Can you assure me that no garden gnomes were harmed in this sequencing? I am calling the PETG (People for the Ethical Treatment of Gnomes) about this one I think.

Re:How is this meaningful?

[heinousjay]

I usually just line the gnomes up by height.

Re:How is this meaningful?

[pyite]

Well, it is generally believed that prime numbers are infinite...

Not sure if you meant twin primes there. It is provable that there are infinitely many primes. Assume that there exists a finite number of primes... p_1, p_2, ... p_n where p_n > ... > p_2 > p_1. Let N = (p_1*p_2*...*p_n)+1. By construction, p_1...p_n do not divide N. Thus, N is either prime itself or divisible by a prime larger than p_n, contradicting the assumption that there are a finite number of primes.

Re:How is this meaningful?

[Sku-Lad]

The more we discover large primes the more we learn about ways to factor them quickly and efficiently.

I don't care how big the prime is, I can factor it quickly and efficiently. And I don't even need a sliderule.

Re:How is this meaningful?

[0rionx]

You're right, my bad. I should have said factoring very large numbers in general, not primes. :P Prime factoring vs. factoring primes.

Re:How is this meaningful?

[Petrushka]

Oh, c'mon, I know the copyright's expired, but giving the attribution for the proof would still be polite :-)

Re:How is this meaningful?

[Kjella]

If the conjecture is false, that is, if there are only finitely many twin primes, certainly the largest pair is important.

Except finding one more pair (or ten, or hundred) doesn't do anything for the theoretical question, because it's possible that there's no twin prime numbers beyond X, where X is far greater than computers can muster. And even if it was the last, you'd have no way of knowing it actually is the last.

Regarding the conjencture, we know there's an infinite number of primes, and we know their statistical distribution. Let's call that probability (for a given value, it's a function) p. Chanches are pretty good twin pairs exist with a probability of about p^2. All you lack is the actual proof.

To show why you need a proof, let's take triple prime numbers. With a probability of about p^3, we should find three primes in a row, right? Wrong, there is exactly one triple prime set (3,5,7). Why? Because x, x+2, x+4 = 0,1,2 mod 3 = one is always divisible by 3.

Re:How is this meaningful?

[Threni]

> Seriously. I'm not a math major, etc. But I'm curious, is this of value? Other than of course as a curiosity.

Are you?

Re:How is this meaningful?

[TropicalCoder]

If there are an infinite number of prime numbers, you would need a much larger search space than 'infinite' to contain them all. Since there can not be more than an infinite number of numbers, therefore the number of primes must be finite.

Re:How is this meaningful?

[jafuser]

However, an important open problem in math is, "Do there exist infinitely many twin primes?" Experts think it's likely enough that the answer is yes

This reminds me of something i've thought about occasionally but is difficult to explain: is there any research into a theory or at least a rule of thumb which states that limits on a pattern tend to be related to the complexity of the pattern?

Another way of putting it: If the definition of a pattern describes a series of numbers without any foreseeable limit, then it seems reasonable to assume that the series continues so long as it is well beyond the "influence" of any constants and scale of iteration (addition, multiplication, factorials, exponentiation, etc.) in the pattern's definition.

For example, if you define a pattern using the number 4, and no part of the pattern has a scale larger than, say, exponential, and if the pattern continues to behave consistently well beyond the "influence" of a magnitude of "4 exponential", then it's probably safe to say that it will continue forever.

I'm guessing there's probably already something to describe this concept, and probably in a lot clearer and simpler terms?

Fun stuff

[A beautiful mind]

Some people are very good at finding these primes. The now disposed record twin prime's finder was prof. Járai, whose lectures I attended.

I find it interesting that the guy who works with insanely cool things like primes gave mind-numblingly boring lectures. He basically read his book out aloud. Some people are just very good at research and very bad at teaching.

Re:Fun stuff

[Mini-Geek]

Some people are very good at finding these primes. The now disposed record twin prime's finder was prof. JÃrai [compalg.elte.hu], whose lectures I attended.

I find it interesting that the guy who works with insanely cool things like primes gave mind-numblingly boring lectures. He basically read his book out aloud. Some people are just very good at research and very bad at teaching.

He didn't have to be good at anything except loading the program that searches for the twin primes on his computer...

Re:Fun stuff

[A beautiful mind]

You do realise that these guys write the algorithms, optimize them to very high levels, then write the code and optimize it? That's lots of work. Without smart thinking you couldn't find such big primes. These are very large numbers. The program runs either on supercomputers or in distributed environments as far as I know.

Re:Fun stuff

[DirePickle]

I find it interesting that the guy who works with insanely cool things like primes gave mind-numblingly boring lectures.

Only on Slashdot.

Re:Fun stuff

[Anonymous Coward]

So, your post amounts to "I saw this guy, he's boring. I like his math, but probably only for the reason that I was once in his presence. I have nothing to contribute on the subject. I probably just skimmed the Wikipedia entry on prime numbers once or twice, and that's about where my knowledge ends" Bravo! We need more fine minds like your own posting here on slashdot.org.

Re:Fun stuff

[fatphil]

Not really.

George Woltmann wrote and optimised code (the FFT).
Paul Jobling wrote and optimised code (the sieve).

No-one else really did anything of any coding significance, as far as I know.

I client-server setup does not count. They're a dime a dozen.

However, it was a good organisational effort, and I support their aims.

Re:Fun stuff

[Tired_Blood]

The now disposed record twin prime's finder was prof. Járai, whose lectures I attended.

That's terrible! He should have at least been recycled!

Or did you mean 'deposed'? :)

Re:Fun stuff

[A beautiful mind]

I was talking about the finding of the now second largest twin prime.

Re:Fun stuff

[fatphil]

OK, thread got detached due to score<1 posts.

Jarai never told the prime number hunting community what tools he uses, so there's no way of judginghow much effort he's put into things. However, an off-the-shelf FFT is within a factor of 2 of the absolute fastest, so in the end all that really matters is how many modern CPUs you can throw at the task. I don't suppose Jarai happens to have access to university labs full of computers, does he?

In other news...

[Anonymous Coward]

Largest waste of supercomputer cycles discovered...

Huh? What?

[Bright Apollo]

So some schlep takes a pair of prime numbers, plops a "2" in the middle of them, and calls this a twin prime? Yeah, I think I'm with Dopey Reply Number One on this, try jamming a a third prime in there and call me when it's done. 350 degrees for twenty minutes.

Oh, wait, my wife tells me the whole number is a prime. Well, that's why she has the Master's in math and I make the money.

-BA

Re:Huh? What?

[tepples]

Actually, a twin prime is a pair of numbers n + 1 and n - 1 such that both are prime. For example, 41 and 43 are twin primes. Incidentally, if n is greater than 4, then n is always a multiple of 6; this is fairly easy to prove to yourself.

I am a math major...

[eklitzke]

I am a math major (although I don't study prime numbers). This is totally, utterly useless, in a practical sense. Well, it might be useful in the field of CS, although I don't know enough about these project to know if any novel algorithms were used. It is sort of interesting though, because the twin prime conjecture (i.e. the statement that there are an infinite number of such pairs) is still unproven, so it's kind of cool to be able to say "Look, we found another pair!"

(On a side note, I don't know of any mathematicians who doubt the validity of the twin prime conjecture. If you proved that the conjecture was false, then you'd be really famous.)

Re:I am a math major...

[Anonymous Coward]

I am a math major (although I don't study prime numbers). This is totally, utterly useless, in a practical sense. Well, it might be useful in the field of CS, although I don't know enough about these project to know if any novel algorithms were used. It is sort of interesting though, because the twin prime conjecture (i.e. the statement that there are an infinite number of such pairs) is still unproven, so it's kind of cool to be able to say "Look, we found another pair!"

One down, infinity more to go. Proof by enumeration, here we come...

Re:I am a math major...

[AKAImBatman]

This is totally, utterly useless, in a practical sense.

Are you kidding me?!? I'm going to use that as my new encryption key! It will be like UBER-secure and take ten hundred billion, billion YEARS to guess!

[...]

Um... I wasn't supposed to tell you that, was I?

Re:I am a math major...

[Anonymous Coward]

Are you kidding me?!? I'm going to use that as my new encryption key! It will be like UBER-secure and take ten hundred billion, billion YEARS to guess!

You misspelled type, but you're right. Since password entry doesn't let you use exponential notation, you're stuck entering a lot of characters every time you want to log in... I'd tell you how many, but my computer isn't powerful enough to calculate log_10(2003663613 * 2^195,000 ± 1)

Re:I am a math major...

[MajroMax]

I'd tell you how many, but my computer isn't powerful enough to calculate log_10(2003663613 * 2^195,000 ± 1)

Because of the properties of log (log(a*b) = log(a) + log(b)), the answer's quite easy: it's approximately 58710.1509793 (continued for a while) ± 1/(2003663613 * 2^195,000)/log_e(10). The ± 1 bit is relatively easy to account for -- a Taylor series expansion of log_e(x + 1) ~= log(x) + 1/x for very large x.

Re:I am a math major...

[TrickiDicki]

That's nothing - I'm going to use it as my password! (starts typing frantically)

Thanks

[JimMcc]

The link was a great short read that made sense of it to me. Thanks for the insight.

Obligatory South Park Reference

[Imexius]

1. Find the largest twin prime numbers 2. 3. PROFIT!!

Re:Obligatory South Park Reference

[Gemini_25_RB]

Trying too hard. (Aside: The generally accepted format would be: 1. Find the largest twin prime numbers 2. ??? 3. PROFIT!!)

Minor correction

[Zadaz]

"The discoverer was a computer in France, owned by Eric Vautier."

I never felt like I should be allowed to take credit for what my screen saver does. Espcially since the whole point is that it does it when I'm not doing anything.

'Course this will all be sorted out when computers can vote.

Re:Minor correction

[RealGrouchy]

'Course this will all be sorted out when computers can vote.

You mean vote for themselves (as opposed to deciding what your vote will be). /tinfoil hat

- RG>

Re:Minor correction

[Kuvter]

Course this will all be sorted out when computers can vote.

I can tell you now, mine votes against DRM.

Re:Minor correction

[malsdavis]

Mine is against the cruel and unethical use of computers in pharmaceutical research.

Re:Minor correction

[wpegden]

I don't know about you, but I'll be making sure my computer votes to give my credit ;)

Re:Minor correction

[huckda]

first of all... did Eric Vautier PROVE this is a twin prime?
and also...

doesn't that +- 1 at the end allow for some sort of variance ?

I'd like to see the proof written out ;) LaTeX that bad boy Eric!

Re:Minor correction

[plopez]

'Computer' is also a job title, albeit archaic. What we seem to have here is a person skilled in computation, who is also a slave. I thought that was illegal or are the French a bit backward?

Good for security.

[r00t]

Now we all know the best numbers to use for a PGP key.

Re:Good for security.

[tloh]

......at least all the smart people who read slashdot. Oh, wait..

I think that was unintentionally funny.

Re:Is this relevant in any way shape or form?

[Anonymous Coward]

It eez "would have been disgraced", you American trash.

ugh

[ILuvRamen]

and to think that the computing power could have been used to find a cure for cancer or aliens (cancer cure yes, cure for aliens no, just finding aliens)

Re:ugh

[dreadclown]

(cancer cure yes, cure for aliens no, just finding aliens)

It is truly sad that no-one cares about the plight of those poor souls infected by aliens.

Re:ugh

[Anonymous Coward]

You've gotta wonder, is it possible to bore cancer to death?

Re:ugh

[Rycross]

You just need to talk to more scientologists for that kind of thing.

MOD PARENT +37 KICKASS

[Anonymous Coward]

Mod parent +37 kickass.

GMP

[bellyjean]

For the curious...

Re:Huh? What?

[XaXXon]

Let's see if it really is fairly easy :)

That gives us 5 other things to try:

No odd numbers can be the base of a twin prime because adding or subtracting one leaves an even number which cannot be prime (except 2), so that knocks out
6n+1, 6n+3, 6n+5.

6n+2 and 6n+4.. why are those no good?

6n+2 doesn't work because 6n is always a multiple of 3, adding 2 and then 1 (for the higher of the potential of the 2 twin primes) is also divisible by three, so it can never be a prime.

6n+4 has the same problem, just on its lower possible twin prime.

That took me longer to figure out that I'm happy with, but I think I got it :)

Learn some English

[l2718]

Actually, a twin prime is a prime number n such that n+2 or n-2 is prime. In everyday English a "twin" is not a pair of people but a member of a pair [under certain assumptions, of course]. The same holds in mathematical English.

Re:Learn some English

[Anonymous Coward]

Fix is easy: "Actually, a twin prime is either of a pair of numbers..."

No Biggest Prime: Proof

[seawall]

> Sometimes I get off the toilet and think I discovered the biggest prime...

Most people here probably know this but:

There is no biggest prime number and the proof is 2 sentences long.... here it is:

Assume there is a largest prime P(n) and thus there is a finite list of all prime numbers: P(1), P(2), P(3),.....P(n). "*" here means multiply.

Well then (P(1)*P(2)*...*P(n))+1 must be prime: whenever you divide that number by prime(s) you always have a 1 left over....but (P(1)*.....*P(n))+1 is obviously bigger than P(n) so our initial assumption of a largest prime number must be wrong. QED.

One of the interesting things for mathematicians (or at least this ex-mathematician) is that you tweak the question just a little bit: "Is there a largest "twin prime"?" and heavy duty brains pound on the question for centuries with no answer. I have had NIGHTMARES over that one....which is one reason I am an ex-mathematician.

Another funny thing about higher math is it has been defended as useless (Hardy: A Mathematicians Apology) but then three guys go and invent RSA and all of a sudden my privacy depends on the properties of prime numbers.

Super Double Hashing Table

[Arakageeta]

Super double hashing table here I come!! As Knuth reccomends: http://www.google.com/search?&q=double+%22hash+tab le%22+%22twin+prime%22+knuth&btnG=Search

Why call them twin primes...

[sehlat]

and not prime mates?

More like who are you kidding?

[Xenographic]

No, one is far more special than being 'merely' prime. One is not a prime number.

It so happens that I have a degree in mathematics, but anyone can just claim that, so I doubt you'll listen to that any more than a Wikipedia link, even if the revision I saw gave the definition of prime numbers correctly.

Re:More like who are you kidding?

[Chacham]

I'm sorry, could you repeat that in Latin?

Re:Huh? What?

[Anonymous Coward]

Prime Twin search Algorithm:

1) For each multiple of 6, test num-1 and num+1 for prime
2) ???
3) Profit! Er, Get Prime Twins!

To quote Fark

[symbolset]

This article is worthless without pictures.

Re:To quote Fark

[john83]

This article is worthless without pictures.

As you wish: The primes. The discoverer and its user.

power consumtions

[AndyST]

It occurs to me that the power consumed for this kind of calculations is quite high. Back when I was doing seti@home, the classic one, they explicitly told people not to let computers running for the sole purpose of calculation, even asking them to turn them of when you guys in the US had a power crisis. There are people running farms of computers just for the fun of it. *sigh*

seti, primes and stuff might be important, but I'd like to still have some power left to radio a reply to E.T.

Prime twins

[DavidV]

...that like to work as a team is what I would find more interesting.

what a coincidence!

[speculatrix]

The pair discovered on January 15th was 2003663613 * 2195,000 ± 1.

what a coincidence! that's the combination to my luggage!

But... aren't all odd numbers prime ?

[dargaud]

Time for an old classic: How to prove that all odd numbers are prime?
Well, this problem has different solutions whether you are a: Mathematician: 3 is prime, 5 is prime, 7 is prime, and by induction we have that all the odd integers are prime. Physicist: 3 is prime, 5 is prime, 7 is prime, 9 is an experimental error... Engineer: 3 is prime, 5 is prime, 7 is prime, 9 is prime... Chemist: 3 is prime, 5 is prime... hey, let's publish! Modern physicist using renormalization: 3 is prime, 5 is prime, 7 is prime, 9 is ... 9/3 is prime, 11 is prime, 13 is prime, 15 is ... 15/3 is prime, 17 is prime, 19 is prime, 21 is ... 21/3 is prime... Quantum Physicist: All numbers are equally prime and non-prime until observed. Professor: 3 is prime, 5 is prime, 7 is prime, and the rest are left as an exercise for the student. Confused Undergraduate: Let p be any prime number larger than 2. Then p is not divisible by 2, so p is odd. QED Measure nontheorist: There are exactly as many odd numbers as primes (Euclid, Cantor), and exactly one even prime (namely 2), so there must be exactly one odd nonprime (namely 1). Cosmologist: 3 is prime, yes it is true.... Computer Scientist: 10 is prime, 11 is prime, 101 is prime... Programmer: 3 is prime, 5 is prime, 7 is prime, 9 will be fixed in the next release, ... C programmer: 03 is prime, 05 is prime, 07 is prime, 09 is really 011 which everyone knows is prime, ... BASIC programmer: What's a prime? COBOL programmer: What's an odd number? Windows programmer: 3 is prime. Wait... Mac programmer: Now why would anyone want to know about that? That's not user friendly. You don't worry about it, we'll take care of it for you. Bill Gates: 1. No one will ever need any more than 3. ZX-81 Computer Programmer: 3 is prime, Out of Memory. Pentium owner: 3 is prime, 5 is prime, 7 is prime, 8.9999978 is prime... GNU programmer: % prime
usage: prime [-nV] [--quiet] [--silent] [--version] [-e script] --catenate --concatenate | c --create | d --diff --compare | r --append | t --list | u --update | x -extract --get [ --atime-preserve ] [ -b, --block-size N ] [ -B, --read-full-blocks ] [ -C, --directory DIR ] [--checkpoint ] [ -f, --file [HOSTNAME:]F ] [ --force-local ] [ -F, --info-script F --new-volume-script F ] [-G, --incremental ] [ -g, --listed-incremental F ] [ -h, --dereference ] [ -i, --ignore-zeros ] [ --ignore-failed-read ] [ -k, --keep-old-files ] [ -K, --starting-file F ] [ -l, --one-file-system ] [ -L, --tape-length N ] [ -m, --modification-time ] [ -M, --multi-volume ] [ -N, --after-date DATE, --newer DATE ] [ -o, --old-archive, --portability ] [ -O, --to-stdout ] [ -p, --same-permissions, --preserve-permissions ] [ -P, --absolute-paths ] [ --preserve ] [ -R, --record-number ] [ [-f script-file] [--expression=script] [--file=script-file] [file...]
prime: you must specify exactly one of the r, c, t, x, or d options
For more information, type "prime --help'' Unix programmer: 3 is prime, 5 is prime, 7 is prime, ...
Segmentation fault, Core dumped. Computer programmer: 3 is prime, 5 is prime, 7 is prime, 9 is prime, 9 is prime, 9 is prime, 9 is ...
Oops, let's try that again:
3 is prime, 5 is prime, 7 is prime, 9 is ... 3 is prime, 5 is prime, 7 is prime, 9 is ... 3 is ...
Um, right. Okay, how about this:
3 is not prime, 5 is not prime, 7 is not prime, 9 is not prim

Re:But... aren't all odd numbers prime ?

[Anonymous Coward]

Minesweeper addict
        1 is green, 2 is blue, 3 is orange, 4 is red...

Actually, 1 is blue, 2 is green, 3 is red, and 4 is purple.

-Minesweeper addict

NO NO NO

[Kjella]

Well then (P(1)*P(2)*...*P(n))+1 must be prime:

No, no and even more no. Let's say my list of known primes is (3,5). 3*5+1 = 16 is not prime, all you've proven is that your list of primes is incomplete. It is only an existance theorem, and can not be used to find new primes.

Re:NO NO NO

[Kjella]

Reread your formulation once again, and you claim you can list all primes less than p(n), which is different than the standard formulation of Euclid (he just says, take a list of known primes). But you're still wrong:

2*3*5*7*11*13=30030
30030+1=59*509

Re:NO NO NO

[eldepeche]

{P(1), P(2), ...} are all the prime numbers, not some list of the ones you know. 2*3*5+1=31, which is prime.

No one said it could be used to find new primes, either. It's a proof that there are infinitely many primes.

Re:NO NO NO

[Anonymous Coward]

I was as confused as Kjella, until I found the euclid's proof myself ( http://www-users.cs.york.ac.uk/susan/cyc/p/primepr f.htm )

I suck at maths, but this is what i made from it:

{P(1)*P(2)*...*P(n)}+1 is either a prime, or at least one of its divisors is a prime larger than P(n).

So i guess that means that there are infinte number of primes then.

Re:NO NO NO

[Orkan]

You're right in a way, that method doesn't give you a prime in general. No one suggested that it did though. The proof was a proof by contradiction, i.e assume notX and generate a contradiction. Based on the assumption that there is a largest prime the procedure works fine. P(i) does not divide P(1)...P(n)+1 for any i in 1..n and by the assumption these are the only primes. The whole body of the proof is showing that "There is a maximum prime P(n)" => "There is a prime > P(n)" giving the contradiction. Therefore there is no maximum prime P(n) and so there must be an infinite number of them.

Re:NO NO NO

[SashaM]

>>Well then (P(1)*P(2)*...*P(n))+1 must be prime:

>No, no and even more no. Let's say my list of known primes is (3,5). 3*5+1 = 16 is not prime,

The entire proof is by contradiction, so the assumption at the point where he says this is that the set of primes is indeed finite. That is of course false, so the consequent can also be false and the statement will still be true. The truth value of "if 3 is even then so is 5" is TRUE.

It is only an existance theorem, and can not be used to find new primes.

Actually, it does give you a way to find new primes, just not a practical one. Suppose you know all the primes less than some number N and they are p1,...,pn and let P=p1*...*pn+1. Now, P, when divided by each of your known primes gives a remainder of 1, so neither of your known primes divide it. Therefore either P is a prime, or it is divisible by a prime between N and P. This gives you a finite (but huge) range of numbers to look for a new prime.

Re:NO NO NO

[Anonymous Coward]

Insightful? Mod parent down -1: wrong. He didn't understand the proof the gp posted. It didn't proport to find new primes. It was an existence proof. Leaving the parent at 3 is like letting a post that intentionally says 2+3=8 stay at 3.

yes yes yes yes

[wpegden]

The poster is not wrong, his proof is correct. It is a proof by contradiction. He ASSUMES (for the sake of trying to find a contradiction), that there are finitely many primes P(1), ..., P(n). If there WERE only those finitely many primes, than the number P(1)*...*P(n)+1 WOULD be prime (because it's not divisible by any of them), which WOULD be a contradiction. Get it? Of course it's true that 2*3*5*7*11*13+1 may not be prime. But the poster didn't prove that P(1)*...*P(n)+1 is ALWAYS prime, he proved it was prime IF those are the only primes, which is enough to get a contradiction.

Anyways, talking about what Euclid did is kind of irrelevant here (except from a historical perspective, of course). What he said wouldn't hold up in most math classes these days. Rather than doing an actual general proof, he says, "assume there are only 3 primes p,q,r. Then p*q*r+1 would also be prime, contradiction!" or something like that. Proof conventions have changed somewhat since then ;)

Anyways, I guess this shows us that Slashdot's moderation system is no substitute for peer review in mathematics, even for really basic problems... surprise!

Re:yes yes yes yes

[Kjella]

Anyways, I guess this shows us that Slashdot's moderation system is no substitute for peer review in mathematics, even for really basic problems... surprise!

I've never seen a variation of that mathemathical proof that says that p_1*...*p_n+1 is prime through some implicit redefiniton of the word "prime" to mean "not divisible by the assumed set of primes", they all tend to point to the prime factorization theorem and merely conclude that the list of primes must be incomplete. I dare you to find me one example of a mathemathical textbook that says "p_1*...*p_n+1 is prime".

And yes, I did study math at the university (it wasn't my major though) so don't go all high and mighty on me about the standard of proofs, I know that normally you formulate that proof starting with a list of known primes, and prove it's incomplete, not start with a highest prime and prove there's one higher which was an odd reformulation, though I suppose both work. Also there are many people that think that formula gives you primes and the original parent only furthers that fallacy.

Re:yes yes yes yes

[wpegden]

The definition of prime is usually: n is prime if n has exactly two positive divisors You can make n be a positive integer, or allow negative integers as well. Let's require it to be positive. Anyways, by this definition (by far the most common one) you conclude that x=p_1*...*p_n+1 is prime, if you assume that p_1,...,p_n are the only primes (since if x had any positive divisors other than 1 and itself, one of the primes would have to divide it, which it can't).

I'm not trying to get all high and mighty! On the contrary, my post was pointing out that the original poster's proof was absolutely correct, even though your response ridiculed it (subject: "NO NO NO" and first line: "NO, NO, and even more NO"). My comment about the moderation system was not a slight at you, but at the fact that the original post was modded lower than your responses.

In any case, there is no difference between assuming there is a finite list of primes and proving it can't be complete and assuming that there is a greatest prime p_n and so all the primes can be named p_1,p_2...,p_n.

Metroids?

[andrewd18]

I'd like to see the largest known twin metroid, but we'll deal with that at another prime. (horrible puns ftw!)

SEPARATED BY 2?

[c0d3r]

Please define what it means to be separated by 2. Does it mean that p1 - p2 = 2 ?

Re:SEPARATED BY 2?

[mdsolar]

No, actually -2 if you are taking them in numerical order.

And once again

[mapkinase]

I am impressed by a human genius.

See this MathWorld page;Re:How is this meaningful?

[Anonymous Coward]

Another great web page to put this discovery in context, with examples, citations, hotlinks, equations, tables, and the like:

Weisstein, Eric W. "Twin Primes." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/TwinPrimes.html

-- Prof. Jonathan Vos Post

All twin primes have the same number of digits

[mdsolar]

Proof:

The average value of a twin prime pair must even or else both members of the pair will be even and not prime.

The only even numbers which could be an average of a twin prime pair and lead to the pair members having different
numbers of digits have the form 100...0 and the number one less than these is of the form 99...9 which is divisible
by 3. QED.

How trivial, yet fun.

I hate to say this...

[nule.org]

but I have to.

We are pentium of borg!
Division is useless!
You will be approximated!

Clarification: Re:NO NO NO

[seawall]

OK, I must be doing this badly.

What we have here is (and this is all it is) a statement that there must be an infinite number of primes.

IF there could be a finite list of all primes (not that I know them) THEN

I can find one more prime with this formula.

Wait a second, that's a contradiction, therefore there can't be a finite list of all primes.

Using that formula to find more primes is the logical equivelent of saying: If I had a purple cow on my head that spit all primes, I could use it to find more primes. I will use purple cow spit to find more primes.

If you want a real proof it should be easy enough to find...usually somewhere before page 50 in any first number theory book. It really is short as such things go and quite a jaw dropper first time it is seen. If you care you deserve seeing it done right.

I apologize for errors in my sketch of a proof, it was wrong to call it a proof.

I need to follow an old theorem here: Seawall should NEVER post a reply to troll in anger at 1am especially wth math.

Proof left as an exercise to reader.

Re:Clarification: Re:NO NO NO

[wpegden]

seawall, your original proof was actually perfectly fine. It's true that you can't use it to find primes by simply computing p_1*p_2*...*p_n+1, but it's a proof by contradiction, so it doesn't matter: you assume what you're trying to prove is false, and use that fact to reach a contradiction. Along the way, you might prove things that are false (since you're using the incorrect assumption that the statement is false). But when you get the contradiction, it still shows that your original assumption had to be wrong, so in fact the statement WAS true. Ughhh... have I made this more confusing or less? Anyways, it's like this: let's say you're trying to prove that there's no purple cow on your head that spits out robots which can't exist in the vicinity of purple cows. (Okay, bad example here, but I'm trying ;) ) Here's a "proof" that there is no such thing on your head. If there WAS, it would spit out the aforementioned robots. But then the robots would exist in the vicinity of the cow, contradiction. Therefore, there can be no such cow on your head. Now apart from being silly and sort of imprecise, this kind of demonstrates the point: here too, you can't use this proof to "find" robots that can't live in the vicinity of purple cows. But the proof is still valid, such cows can't exist, because from there existence you can derive a contradiction. Similarly, a largest prime cannot exist, because from its existence you can derive a contradiction.

Re:Clarification: Re:NO NO NO

[seawall]

Thanks. Actually we appear to be in violent agreement.

If the goal is to be logically correct, I didn't do too bad a job. If the goal is clarity and understanding: well, let's face it: The majority of responses have a subject line of "No No No": pretty miserable!

I confused the heck out of people who are plenty smart enough to understand what I was trying to express; if only I had expressed it better from the beginning. That is a failing and I think it a big one.

Now I've added purple cows for heavens sake and fear only making things worse.

Ah well, on to less controversial things: like emacs vs vi :-).

OK - Here's my contribution...

[TropicalCoder]

My prime number generator algorithm finds all the primes in the 64 bit number space in 100 lines of C. Tried to post this, but I guess it got filtered out by the lameness filter.

Re:Huh? What?

[tepples]

Prime Twin search Algorithm:

1) For each multiple of 6, test num-1 and num+1 for prime
2) ???
3) Profit! Er, Get Prime Twins! Ouch. What's the runtime of that algorithm? When searching for pairs of twin primes, there has to be a better way to choose which multiples of 6 are likely to produce such a pair.

"A006512 Greater of twin primes." at OEIS

[Anonymous Coward]

See also a table of the first 1,000 greater of prime twins, and more references and stuff at:

"A006512 Greater of twin primes."
  at The On-Line Encyclopedia of Integer Sequences
http://www.research.att.com/~njas/sequences/A00651 2

5, 7, 13, 19, 31, 43, 61, 73, 103, 109, 139, 151, 181, 193, 199, 229, 241, 271, 283, 313, 349, 421, 433, 463, 523, 571, 601, 619, 643, 661, 811, 823, 829, 859, 883, 1021, 1033, 1051, 1063, 1093, 1153, 1231, 1279, 1291, 1303, 1321, 1429, 1453, 1483, 1489, 1609, ...

-- Prof. Jonathan Vos Post