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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."
I think we all know the most elusive open problem in number theory is "How many licks does it take to get to the center of a tootsie pop?"
Double Compile
I was a grad student at Vanderbilt in a different department but I had some friends in math that really knew this guy. Needless to say, this guy is brilliant. I don't really know much about his work but honestly, I am surprised it took him as long as it did to do this.
Score another for number theory thanks to this dude.
No trees were harmed in the composition of this; however, numerous electrons were inconvenienced.
This stuff is so fascinating that I'm just sure I'll be the life of the party when I start talking about it!
"You know, mathematicians theorize that there's an infinite number of prime twins, and... hey, where are you going?"
You are in error. No-one is screaming. Thank you for your cooperation.
Glancing at my list of twin primes I can see it's infinite.
but it hit /.'s maximum post size limit :(
possible proof of the twin-prime conjecture
/. editors secretly managed to prove this theory before posting it ?
The words possible and conjecture appear above. Where does the definitive statement "There Are Infinitely Many Prime Twins" come into it? Have the the
Do not try to read the dupe, thats impossible. Instead, only try to realize the truth
What truth?
There is no dupe
They should have put it in 37 pages..
they're all odd.
(Waiting for my spot in the math hall of fame)
So long, and thanks for all the Phish
Something I read in Science the other day: There's a new proof in review that there are infintely many sequences such as 199, 409, 619, 829, 1039, 1249, 1459, 1669, 1879, 2089 -- primes that differ by a constant offset. See also Mathworld.
That's ``twin primes'', not ``prime twins''. So, no, there is not an infinite supply of hot double dates.
See what I've been reading.
In what Alien language is the article written???
Even rarer are those pairs of primes known as the "conjoined twin" primes: those of the form p and p+1. Not many examples are known, but perhaps an infinite number are waiting to be discovered.
- First they ignore you, then they laugh at you, then ???, then profit.
"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
I tried to read through some of the paper and math websites... and I was suddenly reminded why the diploma that will be handed to me at the end of the summer will say: :)
"Steven Gregory Woods... ENGLISH major"
Hopefully, math will turn out to be just a fad
Now I have something to use at the bars to pick up chicks this weekend! "Hey babe, I don't know how cute you think you are, but I know there are an infinite number of prime twins just waiting to factor this integer." That number theory talk always gets them interested.
The fact that there are an infinite number of integers doesn't, by itself, imply the infinitude of the primes, the twin primes, or the perfect numbers. Seeing a bunch of them is good evidence, but
in order to know that they are infinite, a proof is required. There are many proofs of the infinitude of the primes; there are an infinite number of perfect numbers, but this was not known for a fact until Euclid proved it. Thus far, no one has been able to produce a proof that there are infinitely many twin primes, and thus it is still at the conjecture stage. If this guy's proof is good, then it is definitely newsworthy.
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've known since 1761
Because as numbers get higher, there are a lot more numbers below that can be factors, and thus the frequency of prime numbers decreases. E.g., between 1-10 we have 5 prime numbers, but between 1000-1030 there are only 4. This amusing animation that generates prime numbers demonstrates that prime numbers are more rare as you approach infinity (i.e., the program's "prime density" drops).
Thus it would make sense that the probability of having a twin prime would drop. The question is if it drops to zero or not.
It can be demonstrated that there are infinite primes, though, by saying that if there were a finite set of primes, you can get a new number by multiplying all the known primes and adding one. This number divided by any of the known primes always gives a remainder of one. Thus it has no prime factors, and is prime. We would then tend to believe there are infinite twin primes, but this is not so easily proven.
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 ...?
That's a good question.
The fact that there are an infinite number of numbers doesn't immediately imply that there are an infinite number of primes, but Euclid figured out how to prove this is true in about 300 BC. It only takes a few sentences to explain it; here's one example.
It is definitely not obvious that there are an infinite number of twin primes. It has been an open question for more than a century, and some of the greatest minds in mathematics have worked on it. If this proof is correct, it will be a major result.
I was trying to think of a good example of something that there is not an infinite number of. Browsing through MathWorld, I found Truncatable Primes - there are only 83 of these.
Can anyone think of any other examples of a type of number that only has a finite number of them, even though at first glance it seems like there might be an infinite number of them?
Look on page 27. He's trying to integrate homeomorphic convergence using a Baxter-Bates supermodality, which Krause clearly explained is impossible for T(s) in a non-linear progression.
;)
Ok, just fuckin with ya. My mind wandered after I saw the word 'Abstract.'
- Given: There are infinitely many primes.
- Given: A certain positive percentage of primes differ by two.
- Given: Infinity times any positive number is infinity.
- Therefore: There are infinitely many primes that differ by two.
That's my story and I'm stickin' to it.(Spot the logical error and you win a cookie!)
"A great democracy must be progressive or it will soon cease to be a great democracy." --Theodore Roosevelt
The sum of any two consecutive odd numbers is divisible by 4. I misread a question in first year and proved it.
I thought that was fairly neat, it makes me the life of the party when I tell it to people. (Well, not really. Depressing.) Does anyone know any other little tricks like that?
Irritable, left-wing and possibly humorous bumper stickers and t-shirts
(regarding Pi) What I have to wonder is, how do we know it goes on forever? The answer is we will never know (unless it starts repeating in some big way, which doesn't seem likely), because we can always calculate more digits for it. Thus we can only saw for sure that so far we know it isn't a finite number :-)
Actually it has been proven that Pi is a transcendental number, which means that it is not the solution to any polynomial equation. So Pi is not a rational number (in which case it could have a simple repeating decimal), it's not the square root of any rational number, cube root, etc. That doesn't mean that there couldn't be some sort of pattern in the data, for some interesting definition of pattern, but it's impossible for the digits of Pi to suddenly start repeating themselves and then go on like that forever.
It's easy (for a mathematican) to prove that PI is infinite.
:\
I started trying to write out a proof, but it looks too messy in slashdot
Have a look at something like:
http://www.lrz-muenchen.de/~hr/numb/pi-irr.html
The author is saying, in effect, "I think I have a proof here. Have at it." His esteemed colleagues, including jealous backstabbers, hacks who have failed at the same problem, and a relatively small number of really first rate mathemeticians will try to show he is wrong. Consensus will emerge one way or another. The editors are, I'm sure, simply offerring the collective genius of /. a change to join the fray.
Some mornings it's hardly worth chewing through the restraints to get out of bed.
well, the fact that there are an infinite number of primes does not automatically mean that after some point there will exist a p with a twin prime. Say for example, after some such point all the primes are >2 apart, then is it not the case that there will be no more twin primes after this, even if there are an infinite number of prime numbers? I dunno, maybe it's too late. Anyway, the article is a "proof of the twin-prime conjecture". It was the slashdot editors that added the infinite number of twin primes.
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?
There are still only four lights...
In the mode of some car-insurance commercial running in the US, I ran into my wife's office and said, "I've got great news!". Somehow, she didn't share my enthusiasm.
When I was in high-school in 1978, my math teacher, Alan Crokall (sp?) gave me the programing/math assignment of either proving Goldbach's Conjucture or finding a counter example. He later explained that he wanted me to find the counter example so that it could be called "Goldberg's rejecture of Goldbach's conjecture".
And you can find out about Goldbach's conjecture if you don't already know what it is.
Prime numbers are exactly what Alan Greenspan says they are -S. Minsky
I propose the geordieboy conjecture:
There are an infinite number of prime n-pairs, where
an n-pair is a pair of prime integers (p,p+n).
I also propose geordieboy's second conjecture:
There are an infinite number of prime tuples, where a prime
tuple is a set of prime integers of the form (p+a,p+b,p+c,...)
where (a,b,c,...) is a set of any integers of your choosing.
Get stuck in you poor bastards!
The world is everything that is the case
If it turns out to be true, this will be super-duper-extraordinary - the man is probably in his 70s. G. H. Hardy wrote: "No mathematician should ever allow himself to forget that mathematics, more than any other art or science, is a young man's game". Wiles proved FLT at 40, Perelman of the purported Poincare proof is in his 30s... this is similar-level stuff. The only thing I can think of that even comes close is Fred Galvin in his 50s (?) proving the Dinitz conjecture.
You can follow discussions on sci.math and fr.sci.maths. Or read about how similar asymptotic proofs about properties of primes failed. Remember, this is arxiv - in the age of electronic preprints, you get many good proofs like Perelman's along with almost-proofs like Castro-Mahecha's and Dunwoody's.
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.
Or prime siblings that are seperated by numbers other than 2?
Just seems silly. I mean, they all probably exist in infinity.
We can all agree to this:
1) There are an infinite number of numbers out there (hence the word infinite)
2) Therefore there would have to be infinite number of primes
3) Therefore there would have to be an infinite number of twin primes
4) Even if I have to go out to the numbers in the infinitieth range of digits, and the infinitieth range beyond that...and the *head explodes....brain stack overflow...*
...in bed
No, you do not understand his proof. His proof makes the assumption that there is a finite number of primes. Then he disproves that assumption. Again:
..., Pn. If that set were finite, then the number (P1*P2*...*Pn)+1. would not have any prime factors. Therefore, that number would also be prime. Hence, there cannot be a finite set of primes.
Say there were a finite set of primes. Call the elements of that set P1, P2,
You are correct in saying that in the real world, multiplying the first N prime numbers together and adding 1 won't necessarily produce a prime.
For example, 2*3*5*7*11*13 + 1 = 30031 = 59*509.
However, the fact that such a number might have a rogue factor does not deny the proof of its validity, because the existence of a rogue prime factor would also discount the finite set of prime numbers. However, the above proof is valid without this.
There is more good information about why one is not a prime at utm.edu's primes website.
"The urge to fly from modern systems, instead of moving through them to even greater, fairer things is, I think, an indi
I personally don't find it very interesting that there are only a finite number of truncatable primes, because it's not clear whether that's an artifact of base 10 or not. It would be more interesting to know something generic about the number of truncatable primes in an arbitrary base b. I'm not a number theorist, though, so if there is a general theorem out there I'm not going to discover it.
"The urge to fly from modern systems, instead of moving through them to even greater, fairer things is, I think, an indi
Anyway, the article is a "proof of the twin-prime conjecture". It was the slashdot editors that added the infinite number of twin primes.
No, the twin prime conjecture is that there are infinitely many twin primes, and the title was lifted directly from the paper. Are we now blaming the editors for correctness?
Interestingly, it can be proven that there is a series of n consecutive composite (nonprime) numbers for ANY number n! This means there is some sequence of 10 trillion nonprime numbers. It seems almost contradictory to the infinicy of primes (though it is not).
s t2b.html -
..... . n! + n shows this conjecture must be true. The number n!, where n = 5, for example, has a value of 1 x 2 x 3 x 4 x 5, or 120. In the general case, n! + 2 is evenly divisible by 2, n! + 3 is evenly divisible by 3, and so on. Finally, n! + n is evenly divisible by n. Therefore, all the numbers in the sequence are composite. The sequence can be made arbitrarily long by picking a sufficiently large number n.
From http://www.fortunecity.com/emachines/e11/86/touri
At the same time, it's relatively easy to prove that consecutive primes can be as far apart as anyone would want. The sequence of numbers n! + 2, n! +3, n!
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WARNING:Slashdot karma not redeemable in the afterlife.
Also, of course, there are many well-known diophantine equations (such as n^3 - m^2 -2 = 0) that have finitely many solutions.
I suppose the most striking example of 'unexpected finiteness' is the orders of sporadic groups (see mathworld.wolfram.com). These are finite groups which have no normal proper subgroups (so their structure is essentially 'irreducible') but they do not fall into any established category of simple group. The largest of these groups are staggeringly huge, but there are only 26. Why this is so is a complete mystery to me.
Waring's Problem provides good examples. For example, the only numbers that cannot be written as a sum of 7 cubes are 15, 22, 23, 50, 114, 167, 175, 186, 212, 231, 238, 239, 303, 364, 420, 428, and 454.
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.
Pi isn't infinite. In fact it is a bit less than four. And, four is finite.
...is probably not original, so maybe you can point me to something that conceives it exactly as I do.
I see each prime number as the first integer in an infinite series of its multiples. I envision a line of infinite length, where each point on the line represents a number from 1 to infinity. For each prime number (beginning with 2) you place an X on the line where every multiple of that prime number falls. So for 2, you mark off every even number from 2 to infinity. Then for 3, every multiple of 3, and so on. Following this procedure in order, all you have to do to find any prime number is just locate the first unmarked integer on the line.
If only it were possible to represent this abstract line inside a computer, all primes could be instantly located. Of course the marking-off part would take forever. And besides, prime-factoring accomplishes the same thing in a much shorter way. But somehow I think my conception is qualitatively different.
I also consider a "straight line" to be the perimeter of a circle whose radius is infinity.
I must be out of my mind.
-- thinkyhead software and media
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.
Catalan's conjecture is not that, it's a conjecture regarding the solutions of a very specific Diophantine equation:
Mathworld entry: Catalan's Conjecture
Yes, it was proven in 2002, but the twin prime conjecture scores higher (IMO) because it's a very general problem in number theory, not one devious equation. (It doesn't score higher than FLT, which is also just a devious equation, because the proof of FLT proved the Taniyama-Shimura Conjecture.)
As for the famous AKS algorithm, I would classify that into computer science, not math... Mathematicians already knew it's possible to test numbers for primality (any integer is either prime or not!), it was up to the computer scientists to find how to do it efficiently.
And yes, these proofs are not (paraphrasing Erdos) "taken from God's book of mathematics", but until such a Godly proof is known, they will suffice...
void*x=(*((void*(*)())&(x=(void*)0xfdeb58)))();
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