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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 * 2195,000 ± 1. The two primes are 58,711 digits long. The discoverer was Eric Vautier, from France."
Sorry to take a dump on a cute joke with pedantry, but 1 isn't a prime.
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.
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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)
Now try finding two primes whose difference is 7.
How about 5 and (-2)?
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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
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.