Slashdot Mirror
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 :(
It is called "Prime Twins", they look the same!
Good God. They look so god-damned like the same person... I would say to them, "you want ice-cream cone?", both of them say yes. How in the hell?
Honey, I shrunk the Cygwin
I always thought that if God is anywhere to be found, it'll surely have to do something with prime numbers.. these are truly mysterious aspects of our reality.
we discovered a new way to think.
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
I wouldn't say they're mysterious. Interesting, yes.. Hell, all of mathematics is interesting!
What is your penile percentile?
They should have put it in 37 pages..
Isn't math great?
Hopefully this new paper will have some good cryptographic applications.
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.
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.
Actually, 3.
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???
This is probably because math isn't funny, I mean how many comedians do you know who have a PhD in math?
I Am My Own Worst Enemy
While there are infinite amount of numbers, there are more numbers than others.
For an instance, there are more irrational numbers than integers. This has been mathematically proven.
This proof also shows why it is highly unlikely for a mathematics graduate to ever get a date, and is in fact impossible that they get a pretty one.
Just FYI.
Just check how long until the Olsen Twins are legal
Excuse me, I don't mean to impose, but I am the ocean
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.
If all else fails, use induction to prove?
:|
Boy, all those foundations of computer science courses I took are really paying off.
"There is no spoon." - The Matrix
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.
if any of the twins are sexy. ...
Yes, I KNOW no sets of twins are sexy because one is p+2 and the other is p+6, but come on, it's a JOKE people. *sigh*
I always thought that if God is anywhere to be found, it'll surely have to do something with prime numbers..
Hmmmmmmm, I always thought that He would be in Vegas. With His abilities, He would clean up in that town.
Oops....
same guys, but I meant to link to This cartoon ("Mario Twins")
no comment
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.
Let me introduce you to Tom Lehrer
Now let's go poison some pigeons in the park.
One thing that turned me off about math is the insistence in honoring these long dead people. That's focusing more on the discoverer than the discovery. I'm sure a reasonably intelligent person (like most of slashdot readers) could discover "Euclid's" infinitude of primes theorem. We probably would have if we didn't get in class along with the admonition to respect our elders. Take a patent if you want, Arenstorf. But don't insist people centuries from now worship you over the discovery.
-I am an elective eunuch.
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.'
I'm not sure I understand why this is so hard to figure out.
Assuming that there are an infinite number of numbers (always n+1) then doesn't this have to be the case?
"It is seldom that liberty of any kind is lost all at once." -David Hume
- 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
Not Quite...
Either the number your generate is prime or it is divisible by a prime higher than the one you assumed was largest.
If religous zealots don't believe in Evolution, then why are they so worried about bird flu?
Although they are frequently confused, this conjecture has no bearing on so-called "Wonder Twin" primes, in which the p is in the shape of a polar bear and p+2 is in the form of an ice ladder.
Believe me, I'm as surprised by my comment as you are.
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?
Just out of curiousity, is there a practical reason to prove the existance of infinite numbers of twin primes? Or is this purely a matter of curiousity?
8:00 - Sushi bar patrons are staring at 10-12 people drinking sake bombs
9:00 - Sushi bar patrons are wondering what the hell "naive set theory" is and why the hell all my drunken buddies are talking about it
Member of Orkut? Annoyed with spam?
Irritable, left-wing and possibly humorous bumper stickers and t-shirts
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?"
(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.
Tom never got his degree. He was a career undergrad.
Boobies never hurt anyone. - Sherry Glaser.
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
I'm not a physicist, but it's pretty damn obvious that the sun circles the earth. I mean, it comes up on one side and it goes around and it disappears on the other side. What further proof could we need?
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.
>Maybe I'm being an ignorant
... There are an infinite number of primes, thus an infinite number of pairs such that p is prime, but p+2 is not. Thus there are an infinite number of twins that are not 'twin primes'.
Ok then, try this extension to your theory
The number of 'twins that are not twin primes' clearly outnumber the 'twin primes' so that infinity outnumbers the infinite number of 'twin primes.'
Therefore there are zero 'twin primes.'
Chew on that, "genius"
Not really that obvious. For example, even though there
are infinite number of positive integers, no 3 of them satisfy
x^n + y^n = z^n when n > 2.
If numbers themselves are infinite in number (there must be in order for prime twins to be infinite), doesn't it stand to reason that all "types" of numbers are also infinite? So what's the big deal? Am I missing something here?
There are still only four lights...
This is the best thing that has happened to mathematics research since the proof of Fermat's Last Theorem.
void*x=(*((void*(*)())&(x=(void*)0xfdeb58)))();
not...funny...any...more
*elevator music plays*
Just a little dorky computer math nerd trivia.
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.
> Maybe I'm being an ignorant genius, but isn't it just plain obvious that there is an infinite number of primes, thus an infinite number of twin primes. Prove it. That's the point. It's always seemed obvious, but the point is that no one's been able to prove it. Arenstorf did.
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
I never would have expected those things to be related to primes.
No.
- I am a viral sig. Please copy me and help me spread. [strain #2] Thank you
If we can assume that there is an infinite range of real numbers, then any uniform operation on these numbers would yield a potentially infinite number of results. ie: infinite real numbers, but only so many of these are Primes. Any fraction of the infinite, is still the infinite. So, we have infinite Primes. If only some of these are Prime Twins, then still, any fraction of the infinite is still the infinite. Thus infinite Prime Twins.
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
Nice animation of a bear defacating primes. Kinda puts a fresh spin on going for a number two.
Ceci n'est pas une
I know every one said "2" but "2" is true (it is a truism).
Wrong. Declaring Step 2 to be self-evident is simply unjustified hand-waving. Furthermore, it's committing the fallacy of begging the question, as the premise indirectly claims that the conclusion is true.
In either case, there's still another prime that you didn't count.
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
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.)
Pi lies between 3 and 4. So one thing we can say for sure about it is that it is a finite number.
Doesn't it make you feel good to know that our freedoms are protected by politicans, lawyers and journalists.
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!
---
WARNING:Slashdot karma not redeemable in the afterlife.
There's the root of the problem. Nobody creates an idea, or any of its forms. I don't know if its reverence or guilt. It's something like ego that we haven't evolved away from yet.
-I am an elective eunuch.
To prove that the sun circles the earth, you must prove that the earth doesn't spin. To prove this, ignore gravity and prove that a frame of reference fixed to the earth's surface is an inertial frame of reference.
What you just stated, basically, is a conjecture. Something that seems right, intuitively, but isn't proven.
Another example that seems just as "obvious" might be Goldbach's which seems right and has been tested up to very, very large numbers. But it's still not proven.
To most people, the difference is tenuous, but it's there; and sometimes the difference between having a proof or not has critical applications outside pure math.
If anyone were, for instance, able to find a way to easily factor large primes them a great deal of today's best encryption would become moot. So, any additional knowledge about the properties of prime numbers is potentially important.
-- MG
If I read correctly, Brun's theory (and constant) are saying that dual primes become increasingly difficult to find even with an infinite amount of numbers. To me, this sounds like an asymtote, and the only way to describe Pi would be an asymtotic function of primes, right (since primes are unique and non-formulaic)?
I'm not sure if I'm explaining this correctly, but couldn't this shed some light on pi and e.
Don't think that a small group of dedicated individuals can't change the world. It's the only thing that ever has.
We'll be hosting consciousness on a computer soon. Would such a being want to recognize any of these humans you mention?
-I am an elective eunuch.
How is it circular logic?
Given: A certain positive percentage of primes differ by two. There are known twin primes, so the percentage of primes that are twin primes is not zero. If there are a finite number of twin primes then the percentage of primes that are twin primes approaches zero, but that is not strictly equivilent to zero.
If one accepts your "way out" of the circular reasoning charge, i.e. that a finite number of primes doesn't result in a percentage equal to zero, then the conclusion that this percentage times an infinite number of integers implies an infinite number of twin primes becomes a non-sequitur. So, it's either circular, or a non-sequitur.
To make an analogy, your reasoning is similar to the following:
Theorem: There are an infinite number of positive integers less than ten.
Proof: There are known positive integers less than ten, so the percentage of positive integers that are less than ten is not zero. This non-zero percentage multiplied by the infinite number of positive integers yields an infinite number. Thus there are an infinite number of positive integers less than ten.
See the problem?
What has been proven (or allegedly proven) is that there are an infinite number of arbitrarily long prime arithmetic progressions. Arbitrarily long is not the same as infinite, if you've taken any formal math this should be clear to you.
Can someone explain what possible good knowing the answer to this would be?
If you could reason with religious people, there would be no religious people
well that got rid of the damn badgers at least
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.
the function
... + x (x times)
f(x) = x +
is defined only as f:N -> N . One cannot differentiate this function (it's not defined on a Banach space). You try to do this in line 2.
Also, those names are a really easy way of keeping track of the various results. Which is easier to remember 'Proposition 4.13' or 'Swan's Theorem'? Compare 'The generalised result on patching finitely present modules' or 'Quillen's Patching Theorem'?
I remember a pair of twins back from my high school, the Moore twins. Boy, they were prime. They did everything togeatther, everything.
You've gotta be kidding
3.243F6A8885A308D313
You people are so inflexible. These things e v o l v e.
Thanks for the mulligan. I knew it would piss people off, but when you're an old fart like I am, you begin not to care.
Sorry kids but this just chaps my hide
To obtain a times b, one can multiply a times b.
Please write the word multiply ten times and then use it in a sentence.
God DAMN, for ONCE in my life, I post drunk, and everyone gets their panties in a knot.
Teaching you to spell?
Teaching you to think would probably be a good thing too - but I'm not sure a mathematician would be appropriate for that task.
Ok. I'll bite.
Let's say that I've got a *really* big prime. Something, say on the order of 10^10^10^34580.
It's factors are 1 and its self.
There. I've factored it. Do I get a cookie?
This Lawn Supervisor is working on a sprinkler maintenance job, when he starts working on a Findley Sprinkler head with a Langstrom 7 inch wrench.
Well his apprentice leans over and says "Hey, you can't work on a Findley Sprinkler head with a Langstrom 7 inch wrench!"
Well this infuriates the supervisor. So he grabs volume 14 of the Kingsley manual. Which says "The Langstrom 7 inch wrench can be used on the Findley Sprocket".
The apprentice retorts, "I says Sprocket, not Socket!"
Were those plumbers supposed to be here for _this_ show?
Lump lingered last in line for brains, and the ones she got were sorta rotten and insane.
Then there's Abelian groups versus commutative groups. Abelian is elitist and unncessary. If you need to use a name, don't capitalize it.
-I am an elective eunuch.
First I don't really know what ignorant arses are doing here at Slashdot with such ignorant remark such as "Who cares?".
I mean seariously how can you be so thick and short-sighted? Are you the type that embarrass your peers by talking utter crap at meetings? Or the middle-aged IT manager bluffing his way with buzzwords and acronyms in the futile attempt to prove he knows more than the team he selected?
Well, dear uneducated ones, I will tell you who cares: modern science in general, nuclear physics, and most notably cryptography. Mathematics and Number Theory is just a huge pool of knowledge - way beyond our technological time. Many theories which would be classified as useless (by utter idiots) - only triggered huge advances in technology: from chemistry to computer science.
Thank God, we don't have baffoons like those managing what is relevant or not. So if you they want to do contribute something for the good of society - they ought to save the embarrasment and shut up.
If you really want to tell them apart, just have them race each other.
"I think so, Brain, but 'instant karma' always gets so lumpy." - Pinky
"Decepticons FOREVER!!!" - Ravage
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.
Well, there are infinite even perfect numbers iff there are an infinite number of Mersenne primes. There are a few possibilities on how to go about this, but one method I've thought of was showing that there is an infinite 'chain' of Mersenne primes. That is if M(n) is a Mersenne prime, then M(M(n)) is also a Mersenne prime. By induction, it would hold that there are an infinite number of them. Since I don't go by the handle perdos, I have no idea if that would really work.
"Anyone who attempts to generate random numbers by deterministic means is living in a state of sin." -- John von Neumann
If you accept this definition of a circle:
Then you might look at the digital representation of 22/7 as a number that is continually getting closer to infinity. If it repeated it would no longer continue to get closer to the infinite value.
That it falls between two integers means nothing - there exists the same infinite number of numbers between 0 and 1 as between 3 and 4, or 0 and any number.
wrong. even though yer trolling or whatever..
take all the primes up to a point, multiply them together and add 1, and you have a prime number.
3 * 2 + 1 = 7, 2 * 3 * 5 + 1 = 31 ect
We've been so focused on 'stuff that matters' lately
Doesn't mean there aren't errors. There was a conjecture (unproven but believed to be true) that there are an infinite number of twin primes. Now this man believes he has a proof. He's worked through it, and it all seems to fit. However, there is the posibility that he made a mistake. So now it goes out for peer review, and the rest of the community checks his work. If it all checks out, then we have a real proof. If there is an error, back to the drawing board.
Hence a possible proof of a conjecture.
Ok, at some point we'll use a non-spoken numerical language and be much more logical. But for now, there should be some things you can do to promote the discovery over the discoverer. How about base 2 algebra over boolean algebra? Post papers anonymously.
-I am an elective eunuch.
2*3*5*7*11*13+1 = 30031 (not prime)
2*3*5*7*11*13-1 = 30029 (prime)
That would be cool if it worked
Pi isn't infinite. In fact it is a bit less than four. And, four is finite.
2*3*5*7*11*13 + 1 = 59 * 509
Adding one to the product of the primes up to n only guarantees no prime factors less than or equal to n. That's why Euclid's proof works; he assumed that there simply were no larger primes, and showed that his assumption was incorrect.
...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
well you are guarentied a new prime number, be it the number or a factor of the number
I wanted to comment on this, but based on the responses so far, I don't believe there is a single sane person reading all this. It's ok though, It's past my bedtime and I'm still up. Does that make sense?
http://www.chasebrown.com/olson.htm
The same reason that you can have integer values for x^2+y^2=z^2 but put those values to any higher integer power and you'll never find any integers >1 that fit the equation. In regards to the particular story at hand, primes become less common as infinity gets larger, therefore it makes sense that there are an infinite number of primes, but not twin primes, the twin prime conjecture thus makes sense from either viewpoint until someone proves or disproves it.
Regards,
Steve
Perhaps, if it pans out.
However, You're maybe forgetting Preda Mihailescu's proof of Catalan's conjecture and the AKS proof that PRIMES is in P.
(Catalan conjecture is that there are only a finite number of x,y,z,a,b,c \in |N, 1/a+1/b+1/c1 such that x^a+y^b=z^c. It's kind of a generalisation of FLT.)
Both the Catalan proof and this TPC purported proof resort to the use of analysis (integrals, the complex plane) for their proof. This makes them, to some mathematicians, much less elegant. (However, analysis is so powerful that it's used everywhere.)
FP.
Also FatPhil on SoylentNews, id 863
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?
My bank account...
Carbon based humanoid in training.
I'm all for proving the 3n+1 conjecture... Every few months, I get an idea, work at it, and conclude that it doesn't help. Unfortunately, it was shown (by Conway, I think) that similar statements are unprovable. That kind of thing just ruins my day.
If anyone hasn't heard of the problem, pick any integer greater than zero. If it is even, divide by two. If it is odd, multiply by 3 and add 1. Repeat this. The conjecture is that eventually you will get back to one.
Karma: Bad (mostly due to all those "In Soviet Russia" jokes)
There are several different meanings you can apply to the concept of "percentage of an infinite set". For instance, the integers are an infinite set, and I can say quite confidently that 50% of them are even. Of course you will point out, a la Cantor, that there is a one-to-one correspondence between integers and positive integers, and that therefore there are "just as many" so 50% does not make sense. Nevertheless, it is clear what I mean by saying 50% of them are even (although I can't express it less ambiguously at the moment ... it's 5:48 am).
A different example may be clearer. All squares in a plane contain an infinite number of points (cardinality <C>, I believe). But the area of one square can be half the area of another square. So there is a clear meaning to the sentence "{the set of points in square A} is 50% of the size of {the set of points in square B}."
Strange. The first Google hit for Tom Lehrer Ph.D. is this biography, which says "In 1960 he stopped performing and devoted himself to his academic calling, returning to Harvard and earning his Ph.D.".
I'm afraid you've been misinformed. The tree in your link was planted in 1954, although it was "descended from one at Newton's home, Woolsthorpe Manor". (Source: Trinity College, An Historical Sketch, by G. M. Trevelyan, former Master of the College.) It was at Woolsthorpe that the alleged apple incident occurred.
Hell yes, the europeans fought. Especially the germans.
The rest are just "smears."
I am very small, utmostly microscopic.
Simply put: The notion of a percentage breaks down because the denominator in the fraction (by which a "percentage" is defined) is infinite.
HAND.
Anyone who isn't a number theorist, probably. And even then you'd probably need to be looking at analytic number theory not algebraic.
Actually it has been proven that Pi is a transcendental number, which means that it is not the solution to any polynomial equation.
[nit] Nearly.. it's not the root of any polynominal with rational coeffients. If we allowed irrational coeffients then it surely would be! [/nit]
Simon
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)))();
A couple of comments about this.
:).
First of all, I have to say that if history is any guide, this proof is wrong. Incorrect claimed proofs of the twin prime conjecture outnumber correct proofs by a rather large margin
To calibrate this claim a bit it is worth noting that, to date, no one has even been able to prove that there are infinitely many primes pairs within distance o(log n); it has actually been proven for a constant c less than 1, there are finitely many pairs withing distance c log n, which I think is quite dramatic---I don't remember if this requires the Riemann Hypothesis or not.
(By the way, one say that there are infinitely many prime pairs withing distance f(n) of each other if there are infinitely many pairs (a_i, b_i) (both prime) so that |a_i - b_i | max(f(a_i),f(b_i)).)
Twin primes are pairs of primes where both p and p + 2 are prime
Maybe I have gotten something wrong, but does the article really prove that an infinite number of (p, p+2) exist, both prime? I didn't RTFA, because it looked too difficult at first glance, but it appears that they prove something else. Besides, the statement in the heading is a trivial result from the proof 'there is no highest prime number'.
In short: multiply all consecutive prime numbers upto a certain value, call this M. Both M-1 and M+1 are prime and differ by 2.
The article is a bit more involved, and about something else...
Z.
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
A summary of the article appeared in science. The research article is currently under review. but there is a preprint available on arXiv, and also a nice image that shows the result graphically.
I miss my rubber keyboard.(Homepage)
Every prime greater than 3 is expressible in the form 6n+1 or 6n-1, where n is an integer. Easy to prove, but lots of people don't see it at first. For example, 5 = 6(1)-1; 53 = 6(9)-1; 43 = 6(7)+1
etc
An interesting question, and one that I can't answer, unfortunately. Anyone with a little more knowledge of number theory care to comment?
Karma: Contrapositive
No, I understood this perfectly. The statement I had a problem with was
Thus it has no prime factors, and is prime
This is not necessarily true, when you multiply all the factors together it can be composite but since it must have prime factors, they must be larger than the one you assumed was largest.
The original poster states that the number generated is prime which is wrong.
If religous zealots don't believe in Evolution, then why are they so worried about bird flu?
well it means there's another prime number over n. be it the number ittself or a factor of the number
The square root of any integer that is not a perfect square is irrational. View Addenum two of this for details.
Simon.
The error is that in d(x+...+x)/dx (I assume you meant dx, not dt) you treat x as both a variable (of differentiation) and a constant. If you did this on the left side, d(x^2)/dx = x dx/dx = x = 1+...+1. No error that way, really.
I did my senior thesis on (topics strongly related to) the Goldbach Conjecture, and I would be VERY surprised if it isn't proved in my lifetime. A lot of weaker conjectures have been proven--every integer is the sum of at most six primes, every odd integer over 3^(3^15) is the sum of three primes, and every sufficiently large (I forget the known lower limit for this one) even integer is the sum of a prime and a number that is either prime or the product of two primes.
The original Howling Frog is a fictional character and has no UID.
I love your sig. Two nerdy pastimes rolled into one quote.
(Been spending all day at work listening to Live Phish albums, preparing for the great summer show!)
You forget that the assumption stated at the beginning was that the set of primes was finite. Given that, the number created IS prime, which invalidates the initial assumption that the set of prime numbers is finite, and therefore the set of prime numbers is infinite (the count of elements of the set). The original poster was correct with all he said. He only failed to finish the proof with what I stated above.
Our founding fathers removed the guys in charge. Be American. Vote incumbents out.
From: Zbigniew Fiedorowicz
Newsgroups: sci.physics,sci.math
Subject: Re: There Are Infinitely Many Prime Twins
Date: Thu, 03 Jun 2004 13:27:58 -0400
Organization: Ohio State University
Lines: 30
Message-ID:
References:
NNTP-Posting-Host: dhcp-94-53.math.ohio-state.edu
Mime-Version: 1.0
Content-Type: text/plain; charset=us-ascii; format=flowed
Content-Transfer-Encoding: 7bit
X-Trace: charm.magnus.acs.ohio-state.edu 1086283722 19410 140.254.94.53 (3 Jun 2004 17:28:42 GMT)
X-Complaints-To: abuse@osu.edu
NNTP-Posting-Date: 3 Jun 2004 17:28:42 GMT
User-Agent: Mozilla/5.0 (Windows; U; Win 9x 4.90; en-US; rv:1.5) Gecko/20031007
X-Accept-Language: en-us, en
In-Reply-To:
I have heard that Michel Balazard of the University of Bordeaux
has found a serious error in the proof.
>J'ai malheureusement trouv une erreur grave dans l'article d'Arenstorf.
>Le lemme 8, page 35 est manifestement faux, et il est fondamental. Il
>est possible que la dmonstration puisse tre rpare, mais c'est non
>trivial.
The set of point totals a team can not get in football is { 1 }.
paintball