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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.
Glancing at my list of twin primes I can see it's infinite.
but it hit /.'s maximum post size limit :(
"You know, mathematicians theorize that there's an infinite number of prime twins, and... hey, where are you going?"
You would have gotten farther if you had said that without staring the whole time at her "prime twins"
D6 63 0D 70 89 81 BB 8E 7B 7C 5F 5D 54 EA AB 73
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.
Have you never heard of Tom Lehrer? If not, shame on you.
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.'
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?
(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.
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.
It's equally possible that a certain finite number of primes differ by two, not an infinite percentage of primes.
Talking about infinite percentages is meaningless. Think about this question - what percentage of all natural numbers are even? On the one hand, it seems that since every second number is even, there would be 50%, right? But what if I pair each and every natural number to an even number so that two different numbers are paired to different even numbers (a one-to-one map)? Would that mean that 100% of all natural numbers are even? But it is done easily - I would pair each number n to 2*n.
You could try and wiggle out of this problem by defining the infinite percentage to be the limit of the normal percentage until N when N goes to infinity. This would work for some sets, like the even numbers and would even give you a seemingly reasonable answer - 50%. But then consider this question - what percentage of all natural numbers are powers of 2 by this definition? I'll leave that as an exercise to the reader :-)
See Cardinality
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'll explain it like my prof. did :-)
Imagine you arrive at a party and see that some number of men and women are dancing in pairs - each woman is dancing with one man and each man with one woman. You can immediately observe, without counting the actual number of men and women that there is an equal amount of them, right? The same idea is applied to sets (even infinite ones) - if you can pair each element in set A to an element in set B in such a way that each element in B has a pair in A then the two sets have the same "amount" (cardinality is the mathematical term) of elements.
Now, let's take A to be the set of all natural numbers and B to be the set of all even natural numbers. I will then pair each natural number n, to an even number - 2*n. Now, each even number N has a pair - N/2, so we conclude that the "amount" of even numbers equals the "amount" of natural numbers (100% of them, by the naive definition).
You might conclude from this that any two infinite sets have the same "amount" of elements, which seems true at first glance - after all, infinity is infinite, so surely there will be enough elements in any infite set to pair to the elements of another infinite set! This, however, turns out to be wrong. For example, there are "more" real numbers than there are natural numbers. That is, there exists no one-to-one and onto function (Bijection) from the set of natural numbers to the set of real numbers.
To find an infinite number of prime siblings, you first need to find an appropriate set of numbers. To cut down on processing time, you should note that these numbers are seperated by 6, or a multiple thereof.
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.