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There Are Infinitely Many Prime Twins
Posted by
michael
on Fri May 28, 2004 05:49 PM
from the so-the-theory-goes dept.
from the so-the-theory-goes dept.
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."
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Number theory (Score:5, Funny)
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?"
Re:Number theory (Score:4, Funny)
Let's leave the proof to Physics:
One... two... three... *BITE*
............Three.
Parent
Re:Number theory (Score:5, Funny)
Supposedly this guy thought up this new algorithm to calculate large primes in relatively short time. He was granted the use of the university mainframe. He implemented the progam and ran it.
After a couple of days the printer started printing out the number, which was so large it needed a pack of sheets to fit on.
Excited, he looked at the sheets to be gravely disappointed. The last digit was an 8.
Probably an urban legend, but a nice one for sure
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Re:Number theory (Score:4, Interesting)
http://www.sjbaker.org/gallery/lickomatic/index.h
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Re:Number theory (Score:5, Funny)
Put another way, you have entirely failed to receive a wrapper depicting an indian shooting a star.
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One smart dude (Score:5, Informative)
Score another for number theory thanks to this dude.
Re:One smart dude (Score:5, Insightful)
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.
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He makes a mistake... (Score:5, Funny)
Ok, just fuckin with ya. My mind wandered after I saw the word 'Abstract.'
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Re:He makes a mistake... (Score:5, Funny)
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This is why mathematicians are soooo popular. (Score:4, Funny)
"You know, mathematicians theorize that there's an infinite number of prime twins, and... hey, where are you going?"
Re:This is why mathematicians are soooo popular. (Score:5, Funny)
You would have gotten farther if you had said that without staring the whole time at her "prime twins"
Parent
Old news (Score:5, Funny)
I have a better proof (Score:5, Funny)
I have a better proof, and it fits (Score:3, Funny)
- 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!)
Re:I have a better proof, and it fits (Score:4, Insightful)
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.
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Re:I have a better proof, and it fits (Score:5, Informative)
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 [wikipedia.org]
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Re:I have a better proof, and it fits (Score:4, Interesting)
Exactly, which is why that definition is no good either - there is an infinite amount of numbers which are a power of 2, so saying their percentage is 0% makes no sense, or conveys no interesting information. By that definition, an empty, a finite and even an infinite set could be 0% of all natural numbers.
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Re:I have a better proof, and it fits (Score:5, Informative)
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 [wikipedia.org]) from the set of natural numbers to the set of real numbers.
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Re:I have a better proof, and it fits (Score:4, Funny)
You will find your cookie on your hard drive, assuming you're logged in to Slashdot.
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Re:I have a better proof, and it fits (Score:4, Interesting)
Say I have an infinite number of socks. All are white, except 3, which are grey. I have a positive percentage of grey socks, but that doesn't mean anything since that percentage is infinitessimal. It will be infinitessimal for any number of grey socks, so you can't say that you are assumed have a positive percentage of grey socks *unless* you have an infinite number of grey socks, and that's a tautological argument.
Chocolate chip, please
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Can someone buy the editors a dictionary (Score:3, Funny)
The words possible and conjecture appear above. Where does the definitive statement "There Are Infinitely Many Prime Twins" come into it? Have the the
Peer Review (Score:5, Informative)
Parent
38 pages? (Score:5, Funny)
Well, one thing's for sure.. (Score:5, Funny)
(Waiting for my spot in the math hall of fame)
Re:Well, one thing's for sure.. (Score:3, Informative)
Prime Arithmetic Progression also in the news (Score:5, Interesting)
Re:Prime Arithmetic Progression also in the news (Score:5, Informative)
Take it for what it's worth. This stuff is way over my head.
Parent
Calm down, boys ... (Score:5, Funny)
Alien (Score:4, Funny)
twins (Score:5, Funny)
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.
You are either on crack or joking. (Score:3, Informative)
Ugh... math... (Score:3, Funny)
"Steven Gregory Woods... ENGLISH major"
Hopefully, math will turn out to be just a fad
This took 20 years (Score:5, Informative)
Other Number Theory Tricks? (Score:4, Interesting)
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?
Re:Other Number Theory Tricks? (Score:4, Funny)
if a=b, then:
a^2=ab
a^2-b^2=ab-b^2
(a-b)(a+b)=b(a-b)
a+b=b
substitute in the original a=b equation
2a=a
2=1
wtf? So where's the error? :)
Parent
Re:Other Number Theory Tricks? (Score:5, Informative)
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.
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Obvious Generalization (Score:4, Funny)
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!
amazing if it's true (Score:5, Interesting)
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. [temple.edu]
You can follow discussions [google.com] on sci.math and fr.sci.maths. Or read about how similar asymptotic proofs about properties of primes failed. [maa.org] Remember, this is arxiv - in the age of electronic preprints, you get many good proofs like Perelman's [arxiv.org] along with almost-proofs like Castro-Mahecha's [soton.ac.uk] and Dunwoody's. [cnn.com]
Re:cursed mathmaticians (Score:3, Insightful)
It won't. Sorry. Just like AKS, this is something that's entirely in the realm of the theoretical.
FP.
Thank goodness! (Score:3, Funny)
Re:Proof (Score:3, Insightful)
we've known since 1761 (Score:3, Informative)
Re:Proof (Score:5, Informative)
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.
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Re:Proof (Score:4, Interesting)
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
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Lehrer (Score:5, Insightful)
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Re:I didn't RTFA (Score:5, Informative)
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.
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Re:I didn't RTFA (Score:5, Informative)
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 [mathforum.org].
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 [wolfram.com] - 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?
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Re:I didn't RTFA (Score:4, Informative)
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.
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Re:I didn't RTFA (Score:4, Informative)
Waring's Problem [wolfram.com] 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.
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Re:What about prime triplets? (Score:5, Interesting)
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.
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