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."

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?

[Peter Cooper]

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

Re:

[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?

[Record Keeper]

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

Noted.

Re:

[Chacham]

1 isn't a prime

Yes, it is [example.com].

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:

[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:

[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:

[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:

[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:

[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

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:

[sokoban]

] 1 isn't a prime. [wikipedia.org]

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

Re:

[fredmosby]

How about 2 and 5.

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?

[cperciva]

Now try finding two primes whose difference is 7.

How about 5 and (-2)?

MOD PARENT +37 KICKASS

[Anonymous Coward]

Mod parent +37 kickass.

Re:

[DahGhostfacedFiddlah]

Nice [mathforum.org] try [google.ca].

Re:

[cperciva]

Nice [mathforum.org] try [google.ca].

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:

[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:

[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:

[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:

[jhantin]

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

Re:

[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:

[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:

[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:

[Criffer]

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

Re:

[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:

[fatphil]

Bollocks.

2+w and 9+w.

Arse.

Re:

[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:

[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:

[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

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:

[MagicM]

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

*rimshot*

Re:

[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:

[cperciva]

most prime numbers are odd.

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

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:

[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:

[presidentbeef]

Hahahaha! :D
Thanks.

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

[dargaud]

Time for an old classic [gdargaud.net]: 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:

[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 reaso

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:

[hamburger lady]

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

Re:

[JimMcc]

Rrrrriiiiiiiigggggggggggggggghhhhhhhhhhhhhhhhhtttt tttttttttttttttttttt!

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:

[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:

[heinousjay]

I usually just line the gnomes up by height.

Re:How is this meaningful?

[0rionx]

This article [utm.edu] 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.

Thanks

[JimMcc]

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

Re:

[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:

[0rionx]

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

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 a

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:

[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:

[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: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) [wikipedia.org]

Re:

[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

Re:

[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,

Re:

[rifftide]

They're doing it all for their kids.

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

Re:

[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:

[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:

[Petrushka]

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

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 [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.

Re:

[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:

[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:

[A beautiful mind]

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

Re:

[DirePickle]

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

Only on Slashdot.

Obligatory South Park Reference

[Imexius]

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

Re:

[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:

[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:

[Kuvter]

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

I can tell you now, mine votes against DRM.

Re:

[malsdavis]

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

Re:

[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:

[tloh]

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

I think that was unintentionally funny.

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:

[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:

[Rycross]

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

GMP

[bellyjean]

For the curious [swox.com]...

Why call them twin primes...

[sehlat]

and not prime mates?

To quote Fark

[symbolset]

This article is worthless without pictures.

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 [berkeley.edu], 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.

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!

And once again

[mapkinase]

I am impressed by a human genius.

Re:

[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.

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.

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

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:

[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:

[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 contra

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

Re:

[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 examp

Re:

[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 p

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 mor

Re:

[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 sa