## Largest Twin Prime Yet Discovered 160

Posted
by
kdawson

from the Hardy-Littlewood-conjecture dept.

from the Hardy-Littlewood-conjecture dept.

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? (Score:5, Funny)

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

## Re:Are you kidding? (Score:5, Funny)

137

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

## Re:Are you kidding? (Score:5, Informative)

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

## Re:Are you kidding? (Score:5, Funny)

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

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1 isn't a primeYes, it is [example.com].

## More like who are you kidding? (Score:2)

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.

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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

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

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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? (Score:5, Informative)

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.

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## Re:Are you kidding? (Score:5, Funny)

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? (Score:5, Informative)

Now try finding two primes whose difference is 7.How about 5 and (-2)?

## MOD PARENT +37 KICKASS (Score:1, Funny)

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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

psuch that ifpdividesab,pdivides eitheraorb(or both). The integers are an integral domain, and (-5) is a prime.## Re: (Score:2)

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: (Score:3, Informative)

log(x)was base 10, andln(x)was the natural log. Other ways of writing it would be to include the base as a subscript to thelog(), which made it more obvious when doing those tedious exercises to convert the base.## Re: (Score:2)

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.

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lg(x)for base 2.## Re: (Score:2)

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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.

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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.

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

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2+w and 9+w.

Arse.

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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...

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

strike

## Good example of a /. story. (Score:4, Insightful)

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badexample of a*rimshot*

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Nope, it's an

uncommonexample of a## Don't seem too excited (Score:2)

Odd?

## Re:Don't seem too excited (Score:5, Funny)

i'm so sorry.

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most prime numbers are odd.Only on slashdot would the parent get moderated as "informative"...

## Re:Don't seem too excited (Score:5, Funny)

It's the only even prime number.

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Thanks.

## But... aren't all odd numbers prime ? (Score:5, Funny)

How to prove that all odd numbers are prime?Well, this problem has different solutions whether you are a:

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''

Segmentation fault, Core dumped.

Oops, let's try that again:3 is prime, 5 is prime, 7 is prime, 9 is

Um, right. Okay, how about this:3 is not prime, 5 is not prime, 7 is not prime, 9 is not prim

## Re: (Score:3, Insightful)

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? (Score:3, Interesting)

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## Re:How is this meaningful? (Score:4, Funny)

completelydifferent domains. That's the beauty of math.## Re: (Score:1)

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## Re:How is this meaningful? (Score:4, Informative)

## Thanks (Score:1)

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## I am a math major... (Score:2, Informative)

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... (Score:5, Funny)

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

## Re: (Score:3, Funny)

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: (Score:2)

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? (Score:5, Informative)

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: (Score:2)

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

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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 yesThis 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,

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

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Well, it is generally believed that prime numbers are infinite...Not sure if you meant twin primes there. It is

provablethat there are infinitely many primes. Assume that there exists a finite number of primes... p_1, p_2,## Re: (Score:2)

## Fun stuff (Score:3, Interesting)

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.

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Hedidn't have to be good at anything except loading the program that searches for the twin primes on his computer...## Re: (Score:2)

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## Obligatory South Park Reference (Score:1)

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## Minor correction (Score:5, Interesting)

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: (Score:1)

You mean vote

for themselves(as opposed to deciding what your vote will be).- RG>

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Course this will all be sorted out when computers can vote.I can tell you now, mine votes against DRM.

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## Good for security. (Score:5, Funny)

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I think that was unintentionally funny.

## ugh (Score:1)

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

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## GMP (Score:2, Informative)

## Why call them twin primes... (Score:5, Funny)

## To quote Fark (Score:3, Funny)

## power consumtions (Score:2, Interesting)

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! (Score:2)

The pair discovered on January 15th was 2003663613 * 2195,000 ± 1.what a coincidence! that's the combination to my luggage!

## And once again (Score:2)

## Re: (Score:3, Interesting)

Actually, a twin prime is a pair of numbers

n+ 1 andn- 1 such that both are prime. For example, 41 and 43 are twin primes. Incidentally, ifnis greater than 4, thennis always a multiple of 6; this is fairly easy to prove to yourself.## Re:Huh? What? (Score:5, Informative)

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 (Score:2)

## No Biggest Prime: Proof (Score:3, Informative)

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 (Score:3, Insightful)

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

notbe used to find new primes.## Re: (Score:3, Insightful)

2*3*5*7*11*13=30030

30030+1=59*509

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by the assumptionthese 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 (Score:2, Insightful)

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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

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>>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 (Score:2)

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: (Score:2)

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