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There Are Infinitely Many Prime Twins
fustflum writes "R. F. Arenstorf from Vanderbilt University has presented a 38-page possible proof of the twin-prime conjecture using methods from classical analytic number theory. The paper is on arxiv.org and is freely available to the public. Twin primes are pairs of primes where both p and p + 2 are prime. "It is conjectured that there are an infinite number of twin primes ... but proving this remains one of the most elusive open problems in number theory." More information about twin primes can be found on Mathworld."
OK, I'm too lazy to RTFA, but, if there are infinite numbers, why would there not be infinite prime numbers, and infinite prime twins? Are there also not infinite perfect numbers, or ...?
Sorry, why is this news?
Actually, 3.
Slow down!
It's not been reviewed yet.
I'm waiting until Granville, Odlyzko, Mihailescu, or someone similar gives it the thumbs up.
However, it's not obvious tosh, and therefore if it does have flaws it may well be correctible, or at least provide new insight.
The guy certainly _was_ brilliant, but given that he started his peak in the mid-60s, there's no guarantee he's still at it.
FP.
Also FatPhil on SoylentNews, id 863
"Hopefully this new paper will have some good cryptographic applications"
It won't. Sorry. Just like AKS, this is something that's entirely in the realm of the theoretical.
FP.
Also FatPhil on SoylentNews, id 863
Have you never heard of Tom Lehrer? If not, shame on you.
Irrational numbers are mysterious as a whole, I don't think pi is special in that respect. The prrofs are fascinating though. Prooving incommensurability(sp?) takes some very creative thinking.
The darkness... controls the music. The music... controls the soul.
We do know. Look at some of the methods for deriving PI, and it's obvious.
That's like saying "does 8/9 go on forever? How do we know?"
> 2. Given: A certain positive percentage of primes differ by two.
Not necessarily true. It's equally possible that a certain finite number of primes differ by two, not an infinite percentage of primes.
Give me my cookie now.
- For the complete works of Shakespeare: cat
Products of two distinct prime numbers are significantly easier to factor when those primes are "near" each other. Therefore information about how primes are relatively distributed is useful.
Of course, as I said before, this particular result isn't particularly helpful for cryptographic purposes, but you get the idea.
No-one knows what mathematics will be 'applicable' in the future. Who would have thought that the sampling theory of Fourier transforms would become so important in computer image compression?
Well, duh -- (2n-1) + (2n+1) = 4n. ;^)
Here's my neat math trick: Take a multiplication table and go down the diagonal with all the perfect squares. Take one step northeast or southwest on the grid, and the new number is always one less than the one you came from.
The only person I told this to, however, pretty much replied "Well, duh -- (n-1)(n+1) = n^2 - 1."
How is this funny!!?
If no one had confidence in theories, if no one thought it is worthwhile to risk resources to find out something new, we would be living in a pretty fscked up world, don't you think? Isn't that what the essence of 'investment' in research is about?
Instead of a "funny story" or an "urban legend", your story is more like that commercial for Geico where a lawyer taps a prisoner on the shoulder saying "I've got good news." When the prisoner turns to the lawyer with a question full of hopefeul expectation (because the lawyer is the only one who he has at least moderate trust in to get him out of there), he gets an idiotic American smile #6 from him with "I just saved myself a bunch of money by switching to Geico."
I'm sure a lot of people here could derive many of the more famous theorems of math on their own. This is, of course, after they've been educated with hundreds of years of development on those theorems. Euclid didn't have a textbook that fed him all the necessary conditions for his proof and then posed it as a sample problem.
Look at it this way: People have theorized about flying machines for hundreds of years (DaVinci, etc). Any reasonably smart person today can build themselves one, given the proper tools and materials. Does that make the Wright brothers a couple of schmoes who don't deserve any recognition? No, because they were the first to prove that it really could be done, without the benefit of previous research.
And if you'd rather mod me funny than insightful... hell, any reasonably intelligent person today can discover North America without a whole lot of trouble. Doesn't make Columbo's feat any less impressive.
There are an infinite number of numbers, but there aren't an infinite number of pairs of primes p and p+3. (There is obviously only one such pair, 2 and 5.) So it's not trivial that there's not something which prevents there from being any further twin primes.
Two is the only even prime number, which certainly makes it odd.
Um, wouldn't your approach also prove that there are an infinite number of pairs of primes separated by 1 (rather than 2)? Take a course in measure theory and get back to me. (There exist subsets of the integers that are simultaneously infinite and have measure zero.)
Just the fact that you have an infinite set of numbers doesn't mean that anything and everything will be true about those numbers.
The set {2,4,6,...} is infinite, but it only contains one prime.
There's not an infinite number of primes under 10, nor an infinite number of even primes, nor an infinite number of primes equal to 113.
Evan Prodromou | evan@prodromou.name | http://evan.prodromou.name/
It's quite easy to write a program that will verify any proof written out in formal logic.
The problem is that to write out any proof that isn't really obvious anyway in formal logic requires huge amounts of time and space (think 3000+ pages rather than 38, mainly proving the equivalent of 2+2=4).
There are a few people trying to produce a language for mathematics that a computer can understand and check which isn't quite so completely painful and allows you to quote theorems; but they're still quite messy and most of the theorems you might want to use haven't been included yet.
So people go for the time-honoured method of writing proofs in a way that makes sense to a human, and then having people check the logic by hand. Then you need someone who works in the same field to verify it, because people working in different fields won't know the theorems and would have to spend a year or so learning the background.
The reason people don't want to assume something is true until it's been checked is that if you assume that X's proof of a theorem is valid, and you then produce a 200-page proof of the Riemann Hypothesis which assumes the theorem X said he'd proved, then someone checks X's proof and finds a mistake, your proof also collapses.
If it's obvious that there are infinitely many pairs of primes p, p+2, then it's obvious that there are infinitely many pairs of primes p, p+3 for the same reason.
Except there's only one pair like that.
Note also that if pi terminated (i.e. the rest was zeroes) then that still counts as repetition and that would make it rational. Since pi has been proven irrational, it cannot terminate. Therefore, there is no "last digit" of pi.
qntm.org