Slashdot Mirror
Another Breakthrough in Prime Number Theory
Battal Boy writes "From aimath.org: Dan Goldston and his Turkish colleague Yalcin Cem Yildirim have smashed all previous records on the size of small gaps between prime numbers. This work is a major step toward the centuries-old problem of showing that there are infinitely many 'twin primes': prime numbers which differ by 2, such as 11 and 13, 17 and 19, 29 and 31,...I am especially proud of this achievement as Yalcin is a close friend of mine from way back! You may also want to check out the Mercury News Article and Dan Goldston's home page where you can see a photo of Dan's back being slowly but surely broken by two of his children ..." Finding patterns in primes seems to be all the rage.
Any research that can make dealing with prime numbers easier can make cracking RSA asymmetric encryption (the most widely used atm) easier, and thus directly affect your privacy.
Apart from that, of course it's extremely boring, but so is everything, until you think of the applications.
Daniel
Carpe Diem
There's a small joke that goes around in the academic world
Biologists like to think they are chemists. Chemists like to think they are physicists. Physicists like to think they are mathematicians. And mathematicians like to think they are god.
Seriously, think of how much of what we learn boils down to our understanding of numbers, systems, and patterns within them. Mathematics, whether you like it or not, is really a beautiful and elegant subject that very few truly understand.
Even if that had shown that there are an infinite number of twin primes, that would still have absolutely no application to factoring.
At all.
Conversely, having knowledge of more prime numbers also means that new encryption tech can be based on those new theories and enhance privacy. It's kinda a double edge sword...
This has nothing to do with encryption, nothing to do with RSA, nothing to do with practical applications at all. Factoring and cryptography is only a small part of the ocean that is number theory. Please don't automatically assume that anyting about number theory or prime is related to encryption and practical applications. This one certainly isn't. This is about twin primes: the authors have proven that the gaps between consecutive primes are small, asymptotically smaller than the logarithm of the number. This *might* lead to attacks on the twin prime conjecture. Nothing is known yet. This is highly theoretical work. Appreciate pure mathematics, with its beauty, for its own sake.
When prime numbers were first discovered, don't you think everyone initially thought "Great... Who cares about these numbers that have no other factors?" In fact, if I was around when they were first discovered, I bet I would have thought it was a completely meaningless discovery. Who would have thought that later on, prime numbers would become a vital part of encryption? Anyway, my point is that although right now this seems to be of little practical value, who knows what future 'breakthroughs' will be based upon it? Perhaps someone will be inspired by this and come up with a revolutionary way allowing RSA to be cracked in microseconds? And even if no practical use is discovered any time soon, it's still one more thing better understood.
________________________________________________
suwain_2
Strictly speaking, negative numbers can be prime, however, -x is prime if and only if x is prime, so number theorists usually just deal with the positive ones.
As for "counting as factors," one has to look at the definition. An irreducible is a number (well, non-unit element of a ring) such that every factorization has a unit as one of the factors. In the integers, 1 and -1 are the units.
A prime is a number such that if it divides a product, it divides one of the factors. In the integers, these come down to the same thing.
a) the cost to break it should exceed the value of the information it protects
and importantly
b) the time to break it should exceed the useful lifespan of the information it protects
So, hopefully, as the information that has already been transmitted over an insecure network becomes more vulnerable, it *should* also become less valuable because, quite simply, it's becoming old and useless. Ultimately, almost all cryto can be broken given the right amount of time. This is a given, and should be taken into consideration from the start.