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."

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: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.

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: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:Are you kidding?

[cperciva]

Now try finding two primes whose difference is 7.

How about 5 and (-2)?

GMP

[bellyjean]

For the curious [swox.com]...

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 :)

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.

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 [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: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: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: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.