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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 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.
Mathematicians dont tend to produce proof which is machine verifieable ... this remains a possible proof until consensus on it is reached, 38 pages is too much to rely on any single person to verify it's truethfullness.
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.
erm. except one, 2.
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.
2 isn't a twin prime.
Your Mom
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?
Irritable, left-wing and possibly humorous bumper stickers and t-shirts
Take it for what it's worth. This stuff is way over my head.
How can we continue to believe in a just universe and freedom to eat crackers if we have no ale?
(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.
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.
Here's the proof because I'm bored.
.`. the sum of any two consecutive odd numbers is divisible by 4.
Any two consecutive prime numbers can be represented as e + 1 and e + 3 where e is an even number. Their sum is: (e + 1) + (e + 3) = 2e + 4. Because e is even, it's divisible by 2, so let o be e|2.
So 2e + 4 = 2(2o) + 4 = 4o + 4 = 4(o + 1) which is divisible by 4.
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
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
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?
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.
I don't think that holds water. Here are some examples of "types" of numbers that aren't infinite sets:
Clearly, there are some twin prime pairs- 5 and 7, for example. However, just because there are twin primes and there are infinite primes does not mean that there are infinite twin primes. any real number divided by infinite equals 0; or, in other words, the proportion of an infinite population made up by any finite group is 0.
Therefore... one needs to prove independently of the aforementioned observations that there are infinite prime twin pairs; it can never be proved through induction.
Irrational numbers are mysterious as a whole. ...the square root of every prime is irrational.
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.
"So where's the error?"
I'm guessing that's a rhetorical question, but the error is you divide by zero. On line three you are actually are showing 0=0 since anything minus itself is zero and anything times 0 is 0. You then try to divide out (a-b), which is zero, and can't be done.
I can see this fooling people who aren't good at math but probably not math students. It's not like I ever got very far in math, and the problem is easy to spot.
Yeah, it's amazing nobody ever thought of that.
All's true that is mistrusted
Look up "admissible" "k-tuplets" on Professor Caldwell's Prime Pages at http://primepages.org/
FP.
Also FatPhil on SoylentNews, id 863
The sieve of Eratosthenes is the usual name for that idea.
You're wrong to think that it's slower than prime factorisation, though. It may look that way if you play with small numbers, but try writing a sieve program to find all primes up to 10,000,000 and a program that does the same by checking the factorisation of each.
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)))();