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 :(
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 ...?
Sorry, why is this news?
'til we reach a twin prime?
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
A math grad student friend of mine here at Dartmouth came up with several conjectures that would all imply the twin prime conjecture. His work was looking really promising, as I stumbled across some work by other mathematicians that might be invokved to prove his conjecture(s).
Alas, looks like someone beat him to it =). I figured this would happen, though. There're just too many number theorists interested in this problem.
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.
Ha, everyone knows that there are a finite number of primes. And this man calls himself a mathematician!
they're all odd.
(Waiting for my spot in the math hall of fame)
So long, and thanks for all the Phish
I'm just glad he didn't claim that there are an infinite number of prime twins but a finite number of primes.
I think Pi is much more mysterious. I mean think about it, one of the most simple shapes a has a number with an infanite number of seeminly random dicimals in it.
"It is not how things are in the world that is mystical, but that it exists." -Ludwig Wittgenstein
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.
I wish I knew enough about mathematics to make a funny math joke about this. But, alas, I'll have to try and work out an Olsen Twins joke or something equally insipid.
Boobies never hurt anyone. - Sherry Glaser.
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???
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 you haven't read Contact (no seeing the movie doesn't count) then quick go get a copy. 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 :-)
I Am My Own Worst Enemy
Is this kinda like the olsen twins?
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.
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.
For more information read....
(falls asleep)
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.
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
There is an error with b(v) and B(x) on page 22.
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.'
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. Why ? Because there is an infinite number of possible numbers. Sure, primes get farther distanced from each other as the number increases in magnitude, but there's always one around the corner.
I wish mathematicians spent more time on matters of current affair, like gas prices! This kind of 'discovery' bullcrap was fine in the egyptian era where school was nonexistent and everyone was a fricking ignorant unless they happened to suck some royalty's appendage the right way. Today it's obsolete, those who do not understand either don't care, or lack the requisite synaptic ability to function adequately in modern times.
-Billco, Fnarg.com
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
and my D&D character?
- 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
I, for one, welcome our new twin prime masters.
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.
On a related note, the Olsen twins are nearing legal 'adulthood', ahem ;-)
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.
Next at bat: How high is up?
I checked his paper. It's correct.
- AC
Well, I hate to admit this, but I have a major in math among other things, and this paper was absolutely impenetrable. I am totally unable to follow anything - sort of makes me feel stupid. Anyone else in the same boat?
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)))();
headed for the stars and the other stayed here on earth, would they still be prime twins when the first one returned?
/ Wait 'til you hear my "universal conflation" theory.
...the set of primes is infinite. The "guesstimate" would be that pretty much every set based on some criteria (twin, +4, +6 whatever, constant offset) would be, it's just damn hard to prove.
The day you can find some property so that the (non-trivial, e.g. 2+3 being the only ones 1 apart is trivial) set is finite, it's going to be big, real big.
Kjella
Live today, because you never know what tomorrow brings
Wouldn't it stand to reason that if there are an infinite number of NUMBERS, then there would also be an infinite number of anything? Palindromes, prime pairs, numbers that begin and end with "62828312341235", etc?
In other words, every kind of pairing, or category, or oddity you can dream up, there's an infinite quantity of numbers that will match it?
The above explanation is provided just in case the poster was really a dumbass and not a troll as the wise and all-compassionate moderator thought.
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.
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.
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.
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 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.
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.
err subsets of the integers? the integers themselves have measure zero and are infinite (all countable sets have measure zero).
Could you just find a sport to be interested in?
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.
If a=b, then a-b = 0. Can't divide by 0.
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, 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.
Who cares???
Is there anything useful that could possibly come from proving that the number of twin primes is infinite? There are tons of smart people that waste time messing with this stuff that could be ding something benifitial to society. It's sad, really.
You've gotta be kidding
3.243F6A8885A308D313
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.
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.
... is numbers. They're cheap. Dirt cheap. And no mathematician should be able to fool you into thinking otherwise.
Sorry: this is like trying to prove that for every number n there will always be an n + 1 (or n + 2, or...).
I've got better things to do with my time, and I wish other people had better things to do with tax money.
Just imagine if Isaac Newton was sitting there under the tree in Cambridge and the apple fell and good old Isaac said:
'I've got no time now! I'm working on my 38-page paper on prime numbers!'
Yes, that would be very silly. Isaac would never say that. Not in a million light years.
There's namely only one way to do the standing on the shoulders of Giants thing: you've got to find the right Giants, and you've got to have a more down-to-earth footing.
And Giants don't like prime numbers. They hate 'em - with a passion. You can't eat 'em, you can't put 'em in the bank, you can't even bash humans on the head with 'em. Useless, bloody useless they are, like French Citroens with the steering wheel on the wrong side.
I'll take the apples any day. At least they lead to something constructive, to build on. Or correct me if I am wrong: what possible benefit can this prime-ordial nonsense bring?
Besides: Newton didn't get a grant to sit under a tree and let moldy fruit bounce off his noggin.
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.
there are only two questions... are they single and are they hot? wait... what?
Make $5250 with your PayPal Account Guaranteed
But I can prove that there are only a _finite_ number of pairs of primes, where both p and p + 3 are prime.
Film at 11
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
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.
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.
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
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
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
It just struck me that I could run out of problems to solve. /., that there are _no_ mention of Goldbach Conjecture in other topics where math has proven something new.
And it was really a pointy point that poeple started to suggest new conjectures to be proven, when one is.
So why is it fellow
It is like people are somehow throwing new things to the table in an collective manner. How is?
Check out Peter Plichta's Prime Number Cross (Primzahlenkreuz) graphic (towards the bottom of the article).
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)
Sorry, but most USain (as you so cleverly call us) are out enjoying our Memorial Day weekend, this is the weekend where we celabrate saving you sick fuck Euro-weanies from yourselves 50 years ago. Most likely the poster was another "fine" European expressing his Nationalistic views.
Flamebait that may be, but we in europe do remember the debt of gratitude we owe the US. (Perhaps some acknowledgement in the movies that non-americans fought too would be nice though).
And 'usian' is the world's stupidest word.
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}."
where our dictionary was misprinted so that the defintion of the word "competition" was placed under the word "coversation".
If this was an original insight for you, congratulations. But Eratosthenes (B. 273 B.C) gets the credit for publishing first.
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.
The last line is wrong. Each indefinite integral on the second last line has an arbitrary added constant, and there's no reason that the constant must be the same after processes like integrating by parts. So all you can conclude is that 0 = 1 + c (c real), which is true.
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)))();
The sum of a pair of twin primes is divisible by 6.
We all know (I use the term "know" loosely) that pi is a non-repeating, non-terminating decimal number.
My question: How many of the digits are, say, "4"?
Intuition would tell us 'infinite'...but how would you go about proving this? It could be the case that after fifteen quintillion digits suddenly there are no more '4's...
And yes, I think intuition is a key component of mathematics, so if we get different 'feelings' about a number, can't we still get along?
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)
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 :-)
Indeed. Call a number squarefree if it is not divisible by any perfect square (other than 1). Show that the density of the squarefree numbers is 6/(pi^2).
Of course, for certain infinte groups, there are many useful notions of density, and these notions interact in an interesting way. For instance, given a set of integers A, one may take the supremum of |A n I|/|I| over intervals I of length greater than k, and then take the limit as k->oo. This limit is called the upper Banach density of A. Since this is a limit supremum, the limit exists for every set A of integers (although the limit could be 0). Here is a challenging exercise: show that any set of integers with positive upper Banach density contains two numbers which differ by a perfect sqaure.
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
"To make an apple pi truly from scratch, you must first invent the perfectly spherical apple."
Yes there is, as a couple replies to this have pointed out. The mathematical theory that deals with this is called measure theory (http://en.wikipedia.org/wiki/Measure_theory), and it forms the basis of modern probability theory, where percentages of infinite sets are exactly what you want to talk about, and is also very important in analysis, functional analysis and probably a lot of other places. It was invented in part to get a better grip on exactly when and where fourier expansion is a legit thing to do, if I recall correctly. Hope this helps.
22/7 is about as a good a way to generate digits of pi as vegetable shortening. Of course that should be obvious to you, given that being a transcendental it cannot be expressed as the ratio of two integers.
The square root of any integer that is not a perfect square is irrational. View Addenum two of this for details.
Simon.
Call (509) 963-9999.
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!)
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.
... now there is a small disclaimer that reads:
"the proof contained a serious error: namely, Theorem 8 is false".
After all we don't know yet if there are infinitely many twin primes or not!
--federico
The set of point totals a team can not get in football is { 1 }.
paintball