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Twin Prime Proof Erroneous 199

Posted by michael
from the 2+2=5 dept.
mindriot writes "The fairly recent perceived breakthrough in prime number theory regarding twin primes, as mentioned on slashdot, is apparently not quite perfect: 'On April 23rd, Andrew Granville of the Universite de Montreal and K. Soundararajan of the University of Michigan found a technical difficulty buried in one of the arguments in the preprint of Goldston and Yildrim. The main issue is that some quantities which were believed to be small error terms are actually the same order of magnitude as the main term. For now this difficulty remains unresolved.' A more detailed technical description is also available."
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Twin Prime Proof Erroneous

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  • by Dominic_Mazzoni (125164) * on Saturday May 31, 2003 @02:35PM (#6085653) Homepage
    The last paragraph of the "more detailed technical description" is interesting (shown here in LaTeX notation):

    The consensus is that the definition of $\gamma_R$ needs to be changed so that terms like this one do not appear. However, it is not obvious how to do this change. Work is continuing by Goldston and Yildirim and others to rectify the problem. It does seem reasonable to believe that an improvement on the current world record for small gaps between primes will be achieved by these methods; however, the more dramatic result $p_{n+1} - p_n < (\log n)^\alpha$ for some $\alpha < 1$ seems less likely.

    Unless I'm misunderstanding something, it would be more clear if they said that the inequality above holds for infinitely many $n$, because it certainly couldn't hold for all $n$.

    Essentially they're claiming that it's less likely now that the twin prime conjecture will ever be proved using this method, but there's still a pretty reasonable chance that the proof will result in something along the lines that there are infinitely many pairs of consecutive primes that differ only by x, where x is not quite as small as 2 (which is what the twin primes conjecture says) but x is smaller than any value of x that was previously proven. Which would be cool, but nothing to open champagne over.
  • by schematix (533634) * on Saturday May 31, 2003 @02:35PM (#6085654) Homepage
    heard this in an engineering class the other day... What's the contour integral around Western Europe? A: Zero, because all the Poles are in Eastern Europe!
    • A worse math joke - Why did the mathematician name is dog cauchy ?

      Because he left his residue at every pole

      Ducks :P
    • by Dthoma (593797) on Saturday May 31, 2003 @03:17PM (#6085874) Journal
      Q: What did the constipated mathematician do?
      A: He worked it out with a pencil!

      Q: What's purple and commutes?
      A: An Abelian grape.

      Q: Why do you never hear the number 288 on television?
      A: It's two gross.

      Q: What do you get when you cross a mosquito with a rock climber?
      A: Nothing. You can't cross a vector and a scalar.

      Q. How many mathematicians does it take to change a lightbulb?
      A. 1, he gives the lightbulb to 3 engineers, thus reducing the problem to a previously solved joke.

      Q: What's big, grey, and proves the uncountability of the reals?
      A: Cantor's diagonal elephant.

      Q: What's yellow and equivalent to the Axiom of Choice?
      A: Zorn's Lemon.

      Q: What's yellow, normed, and complete?
      A: A Bananach space.

      Q: What is very old, used by farmers, and obeys the fundamental theorem of arithmetic?
      A: An antique tractorisation domain.

      Q: What is hallucinogenic and exists for every group with order divisible by p^k?
      A: A psilocybin p-subgroup.

      Q: What is often used by Canadians to help solve certain differential equations?
      A: the Lacrosse transform.

      Q: What is clear and used by trendy sophisticated engineers to solve other differential equations?
      A: The Perrier transform.

      Q: Who knows everything there is to be known about vector analysis?
      A: The Oracle of del phi!

      =======

      Halfway through a recent airplane flight from Warsaw to New York, there was nearly a major disaster when the flight crew got sick from eating the fish. After they had passed out, one of the flight attendants asked over the intercom if there were any pilots in the cabin.

      An elderly gentleman, who had flown a bit in the war, raised his hand and was rushed into the cockpit of the 747. When he got there, took the seat, and saw all the displays and controls, he realized he was in over his head. He told the flight attendant that he didn't think he could fly this plane. When asked why not, he replied,

      "I am just a simple Pole in a complex plane"

      So, they just had to rely on the method of steepest descents.

      =======

      You know that during the Great Flood, Noah brought along two of every species for reproductive purposes. Well, after a few weeks on the ark, all the couples were getting along fine, except for these two snakes. Day and night, Noah worried that this was going to mean the end of this species.

      Finally when the flood ended and the ark hit ground, the two snakes darted out of the ship and headed to the nearest picnic table where they started to "go at it". It was then that Noah realized that...

      Adders can't multiply without their log tables.
      • I'm a CS student in my second semester, and I only got two or three of those. Guess I'll have to pay more attention my analysis classes. ;)
      • This is probably very inexact, it's from memory and translated from spanish:

        A man had to take a plane but he was very nervous thinking there might be somebody with a bomb on board. He went to the pilot and asked how likely it was that there was somebody with a bomb on board.

        The pilot answered "Well, I wouldn't worry about it at all. It's very unlikely. Probably something like 1 chance in a million". The man feeling somewhat better then asked: "And what is the chance of that there are two bombs on board?".
        • A priest, a physicist and a mathematician are trapped in a burning hotel. The only way out is to jump out of the window and try to land in the nearby pool. So, the priest starts first, he makes a last prayer, jumps and hits the pool. Saved. Next comes the physicist, he looks down, does a short calculation in his head, jumps and is saved. Last comes the mathematician, he looks down, and remains at the window calculating desperately. After a couple of minutes, he jumps, too. However, he flies off into the sky
      • by Dominic_Mazzoni (125164) * on Saturday May 31, 2003 @06:32PM (#6086834) Homepage
        I know this is incredibly nerdy, but it sounds like some people would appreciate it if the jokes were explained to them...

        Q: What did the constipated mathematician do?
        A: He worked it out with a pencil!


        OK, not going to try to explain this one.

        Q: What's purple and commutes?
        A: An Abelian grape.


        A group is a set of things (think "numbers", but they could be sides of a cube, or colors, or anything you want) along with an operation defined on them (like addition or multiplication, but it doesn't have to work like those). When the operation on the group happens to be commutative (like 2+4 = 4+2), the group is called Abelian [wolfram.com]

        Q: Why do you never hear the number 288 on television?
        A: It's two gross.


        A "gross" is a dozen dozen, or 144. Not a very mathematical joke.

        Q: What do you get when you cross a mosquito with a rock climber?
        A: Nothing. You can't cross a vector and a scalar.


        The joke is referring to a Cross Product [wolfram.com], an operation defined on two vectors. You can't take the cross-product of a vector and a scalar.

        Q. How many mathematicians does it take to change a lightbulb?
        A. 1, he gives the lightbulb to 3 engineers, thus reducing the problem to a previously solved joke.


        When a mathematician needs to prove that A implies B, they may instead prove that A implies C where "C implies B" was already proved by someone else, or in a previous theorem.

        Q: What's big, grey, and proves the uncountability of the reals?
        A: Cantor's diagonal elephant.


        The joke is referring to the Cantor Diagonal Argument [wolfram.com], a proof technique that Cantor originally used to prove that even if you tried to associate one real number with every integer, there'd still be real numbers left over. (Amazingly, you can "count" the rational numbers - i.e. all of the possible fractional numbers. As a math major to show you sometime, it's a neat trick.)

        Q: What's yellow and equivalent to the Axiom of Choice?
        A: Zorn's Lemon.


        Zorn's Lemma [wolfram.com] is a mathematical statement which turns out to be true if the Axiom of Choice is assumed to be true, or false if the Axiom of Choice is assumed to be false.

        Q: What's yellow, normed, and complete?
        A: A Bananach space.


        A is space (a set of numbers with a lot of useful operations defined on them) that has a normalization operator defined, and is "complete", which means that the limits of all sequences you can define using numbers in the space are also in the space.

        Q: What is very old, used by farmers, and obeys the fundamental theorem of arithmetic?
        A: An antique tractorisation domain.


        Q: What is hallucinogenic and exists for every group with order divisible by p^k?
        A: A psilocybin p-subgroup.


        A Sylow p-Subgroup [wolfram.com] is a certain type of subgroup (see the definition of a group above).

        Q: What is often used by Canadians to help solve certain differential equations?
        A: the Lacrosse transform.


        The is a technique that makes certain differential equations a lot easier to solve - essentially you take a complicated D.E., substitute certain things in place of any derivatives you see by looking them up in a table, then solve the resulting equation using normal algebra, and finally transform it back also by looking up things in a table.

        Q: What is clear and used by trendy sophisticated engineers to solve other differential equations?
        A: The Perrier transform.


        The Fourier Transform is also used in signal processing, including sound analysis and sound compression algorithms like MP3 and Ogg Vorbis.

        Q:
        • by Schreck (137216) on Saturday May 31, 2003 @09:01PM (#6087389)

          Q: What is often used by Canadians to help solve certain differential equations?
          A: the Lacrosse transform.


          The is a technique that makes certain differential equations a lot easier to solve - essentially you take a complicated D.E., substitute certain things in place of any derivatives you see by looking them up in a table, then solve the resulting equation using normal algebra, and finally transform it back also by looking up things in a table.


          The joke is referring to the Laplace transform. There is no Lacrosse transform.


          Q: Who knows everything there is to be known about vector analysis?
          A: The Oracle of del phi!


          Hmmmm, I don't get this one. Sorry. Anyone?


          The del operator is fundamental in vector calculus. You can define the gradient, curl, divergence and the Laplacian with it. It's also known as nabla.


          So, they just had to rely on the method of steepest descents.

          A way to find the nearest local minimum of a function - works whenever the function is smooth near that minimum.


          No. You're talking about the gradient descent method. The method of steepest descent is a way to find the asymptotic series of a function. I know Weisstein's Mathworld [wolfram.com] agrees with you, but check their references on that page. Arfken and Morse, Feshbach agree with me! I know because I've been studying those two books on this very subject the whole evening before I checked Slashdot. I was mightily surprised to see the method's name mentioned here, believe me.
        • Q: What is very old, used by farmers, and obeys the fundamental theorem of arithmetic?
          A: An antique tractorisation domain.


          You missed this one. The joke is referring to unique factorisation domains. For example, the integers is a unique factorisation domain. 12=2*2*3, 15=3*5... You know what it means. Of course, changing the order of the factors doesn't count as a different factorisation.
        • "I am just a simple Pole in a complex plane"

          Given a complex function, a pole is just a point where the function is not defined (usually because something goes to infinity).

          Sorry to reply again, but this one still needs clearing. Pole is always a singularity. In fact an analytic function may have two different kinds of singularities: poles and essential singularities. Poles are those singularities that can be removed by multiplying the function with a polynomial of a high enough degree. For example, 1/

        • Q: Why do you never hear the number 288 on television? A: It's two gross. A "gross" is a dozen dozen, or 144. Not a very mathematical joke. You're being too intelligent for this part of the joke. It's supposed to be read literally two ate eight Get it? A person named "two" "ate" a person named eight.
        • Just to be extra pedantic. . .

          Q: What do you get when you cross a mosquito with a rock climber?
          A: Nothing. You can't cross a vector and a scalar.

          The joke is referring to a Cross Product, an operation defined on two vectors. You can't take the cross-product of a vector and a scalar.
          The cross product is in fact only defined for pairs of (real, i think) 3-vectors.
      • by isomeme (177414) <cdberry@gmail.com> on Saturday May 31, 2003 @06:34PM (#6086840) Homepage Journal
        "I am just a simple Pole in a complex plane"
        Good thing he took the copilot's seat; the system becomes unstable if there's a Pole in the left half-plane.
      • An (American) Indian chief has three wives, all of them pregnant. One day, the village shaman prophesizes that all three women will not only give birth on the same day, but within the same hour! He then goes on to prepare a teepee with three animal skins: a deer, a bear, and a hippopotamus.

        Well, the special day finally arrives when all three women go into labor. The shaman directs them to go to the specially prepared tent and each lie down on one of the animal skins.

        The woman on the deer skin gives bir
      • Q: What's purple and commutes? A: An Abelian grape.

        Q: What's purple, commutes, and has $N$ worshippers?

        A: A finitely-venerated abelian grape.

        Moooooooooooahahahahah

    • Three statisticians go hunting.

      They see a prize buck.

      One statistician fires... Bang... 10 meters to the left.

      The second statistician fires... Bang... 10 meters to the right.

      The third statistician jumps up and down, yelling "We Got It!".
  • Error? (Score:1, Funny)

    by Anonymous Coward
    Damn, I forgot to carry the 1...
  • by rock_climbing_guy (630276) on Saturday May 31, 2003 @02:47PM (#6085711) Journal
    I have no idea how this proof works because the server melted already.
  • by AtomicX (616545) on Saturday May 31, 2003 @02:47PM (#6085714)
    Their webserver it seems, " is apparently not quite perfect:" It has already been /.ed Obviously evidence of a conspiracy to cover up the mistakes in the theorem. Sssshhh!
  • by CTalkobt (81900) on Saturday May 31, 2003 @02:51PM (#6085740) Homepage
    but the space that I'm allowed to type in here is too short.. :-)
  • by cheezus (95036) on Saturday May 31, 2003 @02:56PM (#6085771) Homepage
    To think you solved something like that, and to be ready to publish, after all that hard work.... then...... oops. guess that doesns't work

    man. i feel sorry for those guys
  • mirror (Score:5, Informative)

    by jroysdon (201893) on Saturday May 31, 2003 @02:57PM (#6085778) Homepage
    aimath.org/primegaps/ [roysdon.net]
    aimath.org/primegaps/residueerror/ [roysdon.net]

    I'm still working on mirroring all 47 images, but the text is there, and the img tags have great alt text descriptions.
  • I see a lot about these sorts of massive mathematical problems. I can understand calculating pi to the nth point as it is used in calculations, but what sort of benefit does mankind get from working out twin primes? In fact, do primes do anything for us anyway?

    I'm well aware of what primes are, I just have never found a use for them!
    • Yes considering a lot of our encryption is based on prime numbers. So you figure a simple way to get around it and you make a lot of encryption outdated and useless. So yes, it is important.
    • by Daniel Dvorkin (106857) on Saturday May 31, 2003 @03:10PM (#6085847) Homepage Journal
      Well, pretty much all current cryptography techniques depend on primes. Whether knowing anything about the occurrence of twin primes has any bearing on crypto, I have no idea.

      The longer answer to your question is: who the hell knows? One of the fascinating things about math is how results that seem utterly abstract when they're [invented | discovered] (not going to get into that argument right now) turn out to have profound applications years or decades or even centuries down the road. Linear algebra was an interesting but rather small and not terribly important field of study before computers came along ...

      The twin prime problem may remain a curiosity of number theory forever, or it may turn out to be fundamental to some new application that's just down the road; there's no way to know. But given the history of math's progress from pure theory to the basis of technology we use every day, I'm betting on the latter.
      • Whether knowing anything about the occurrence of twin primes has any bearing on crypto, I have no idea.

        I have very little knowledge of primes and cryptography, but I do know that the holy grail is the search for larger primes.

        Could the importance of twin primes be that if a corrolary is found that will allow one to predict higher prime numbers because of the n that separates them, it would then become easier to "discover" larger and larger prime numbers?

        • I do know that the holy grail is the search for larger primes.

          Actually, finding large primes is pretty easy [wolfram.com]. Taking a large number and finding its prime factors is not. This conjecture/proof doesn't seem to have any immediate bearing on cryptography.

          TTFN
      • Number theorists have proven that there exists no polynomial function f(x) such that f(x)={primes}. There is, however, a vast literature concerned with the distribution of primes. For instance: Prime Number Theorem: "the prime number theorem gives an asymptotic form for the prime counting function pi(N), which counts the number of primes less than some integer n." Bertrand's Postulate: If n > 3, there is always at least one prime p such that n is less than p which is less than 2n-2. Wilson's Theore
      • What I find amusing is that modern cryptography, the primary application of number theory, is essentially about creating hard problems.
    • by rock_climbing_guy (630276) on Saturday May 31, 2003 @03:12PM (#6085856) Journal
      One really good example of what prime number theory is good for is cryptography.

      For example, in mathematics, it is a well-known fact that it is an easy problem to multiply two numbers. It is a very hard problem to take a number and factor it into the numbers that were multiplied to get the number, especially if it is a very large number.

      If we multiply two very large prime numbers, the result is a very large number that is very difficult to factor; when it is factored, the result will be that it factors only into the original two very large prime numbers.

      Prime numbers also have application in the idea of 'remote coin flipping.' ie. Using prime number theory, it is in theory possible for me to do the equivalent of flipping a coin and you having to guess if it's heads or tails.

      If you still don't understand, consider this. Which is easier to do:
      Multiply 13*17*19*29*57*91*43
      --or--
      Factor 27159925611 into it's prime factors.

      If you can find an easy way to do the second problem, you just might find yourself considered a threat to national security.

    • by HeghmoH (13204) on Saturday May 31, 2003 @03:16PM (#6085871) Homepage Journal
      Pi accurate to about forty-some digits would be accurate enough to calculate the circumference of a circle the size of the visible universe with an error the size of a proton.

      How, exactly, is calculating billions of digits of pi useful, again?

      On the other hand, primes are used for all kinds of good stuff, such as protecting your credit card numbers from evil people. Your conceptions seem backwards.
      • 1. To look for patterns (See Carl Sagan/Contact)

        2. Test out new computer hardware/software

        3. The thrill of the chase. Some people climb mountains, other people calculate billions of digits of Pi.

        • 1. To look for patterns (See Carl Sagan/Contact)
          Right.... You'd be more productive feeding the Bible into a random number generator.

          2. Test out new computer hardware/software
          This is not useful. There are better ways to test hardware/software.

          3. The thrill of the chase. Some people climb mountains, other people calculate billions of digits of Pi.
          This is not useful.

          If you're going to reply to posts, you should try to be more relevant to what they say. I never said calculating billions of digits was bo
    • by wass (72082) on Saturday May 31, 2003 @03:26PM (#6085917)
      I can understand calculating pi to the nth point as it is used in calculations

      Even the most precise calculations don't need that many digits of pi. It's amazing how fast orders of magnitude build up.

      Take this extreme example. Suppose you know the radius of the galaxy (define the radius going out to the galactice halo, for instance) to arbitrary precision and your calculation of the circumference is limited only by the precision of pi. If you want to know the circumference town to 10^-15 meters (ie, about the size of an atomic nucleus). How many digits of pi are sufficient?

      The radius of the Milky Way galaxy out to the galactic halo is about 65,000 light years, or about 6e20 meters. Only 36 digits of pi would be necessary!!! And this extreme example is of many orders of magnitude larger than precisions of anything that can be calculated in laboratories today. In actuality, one wouldn't really need more then 12-15 digits of pi, if even that much.

    • First off, do we really need a practical use? Why do we study literature or art? Do we need practical uses for those things too?

      Nevertheless, prime numbers are important for cryptography. This has to do with factoring a number into prime powers. Any positive integer can be written uniquely as a product of prime powers. If I give you two primes such as 6421 and 7873, it's easy to just multiply them and get 50552543. However, given 50552543 it's not at all obvious that the only way to write this as a product
    • Parent poster wrote:

      what sort of benefit does mankind get from working out twin primes? In fact, do primes do anything for us anyway?

      With all due respect, who cares? What sort of benefit does the mankind get from the Mona Lisa? From NASCAR? From owning pets? Need I go on . . . . ?

      Why does everything have to be for some benefit? Most math doesn't have an application, not now and not ever. We're doing more math everyday than we'll ever find uses for. Mostly it is an aesthetic pursuit. People

    • by phliar (87116) on Saturday May 31, 2003 @06:37PM (#6086855) Homepage
      I can understand calculating pi to the nth point as it is used in calculations...
      That is not a very good reason! Let's say we use pi = 355/113 -- an approximation that's been known for many centuries -- to calculate the circumference of the earth. Using that value of pi our estimate will be off by about 30 feet (about 0.00003%). Even 22/7 is only off by 0.1%.

      No; we calculate the umpty-bazillionth digit of pi for the same reason Mallory wanted to climb Everest: because it's there -- and there's cool shit to see along the way.

    • I consider math at this level to be more like art and less like engineering.

      You do it because the problem is beautiful and the solution is likely to be beautiful. If you are lucky the solution will turn out to be not only beautiful but also make a statement about life (or some aspect of life).
  • by Anonymous Coward on Saturday May 31, 2003 @03:10PM (#6085842)
    This story doesn't have anything to do with SCO! Come on, where's today's SCO story? This isn't funny, man, I need my fix!
  • by MrRage (677798) on Saturday May 31, 2003 @03:35PM (#6085955) Homepage
    What would be really important is to prove the Reimann Hypothesis. That would tell us a lot about the distribution of primes.
    • What would be really important is to prove the Reimann Hypothesis. That would tell us a lot about the distribution of primes.

      Actually that's the Riemann hypothesis, but your mistake seems to be a common misspelling, so don't feel too bad.

      It would be nice to know that the Riemann Hypothesis is true. You see the prime counting function pi(x) (= number of primes less than or equal to x) can be approximated with the integral Li(x) = integral from 0 to x of 1/log(t) dt. The Riemann hypothesis is equivalent to
  • sorry, slightly off topic.. but after having a bit of a look around here i've concluded that the most popular phrase on slashdot is 'order of magnitude' :x
    • Re:cough (Score:3, Funny)

      by Paradise Pete (33184)
      i've concluded that the most popular phrase on slashdot is 'order of magnitude'

      Yeah, and it's not even close. "order of magnitude" is more popular by a... heck of a lot. (heh.)

    • by fava (513118)
      Actually the most popular phrase on slashdot is:
      n : ???
      (n+1) : Profit!
  • by drdale (677421) on Saturday May 31, 2003 @04:34PM (#6086258)
    Remember that an error was found when the British mathematician first announced that he had a proof of Fermat's Theorem a few years ago. He was able to fix it, however, and AFAIK his proof is currently considered sound (albeit LONG).
  • by Ridge (37884) on Saturday May 31, 2003 @04:42PM (#6086297)
    It was in here [kraftfoods.com].

    Unfortunately, I devoured it. Damn you Bill Cosby!
  • I still wouldn't mind these guys helping me understand my calculus homework...
  • from the 2+2=5 dept

    Step 1: -1/1 = 1/-1
    Step 2: Taking the square root of both sides:
    Step 3: Simplifying:
    Step 4: In other words, i/1 = 1/i.
    Step 5: Therefore, i / 2 = 1 / (2i),
    Step 6: i/2 + 3/(2i) = 1/(2i) + 3/(2i),
    Step 7: i (i/2 + 3/(2i) ) = i ( 1/(2i) + 3/(2i) ),
    Step 8: ,
    Step 9: (-1)/2 + 3/2 = 1/2 + 3/2,
    Step 10: and this shows that 1=2.

    therefore 2+2=1+1=3; QED

    Thanks for the proof [toronto.edu]
    • In other news, 0.999 repeating = 1.0

      1/9 = .11111111 repeating

      * 9 *9

      ---- ----------

      9/9 = .99999999999 repeating

      1 = .99999 repeating

      Granted, the parent post isn't true while mine is but this is cool to show to people who aren't that math knowledgable. Had some guy looking at it for 2 hours trying to find the problem in it.

  • Damn it! (Score:4, Funny)

    by 1nv4d3r (642775) on Saturday May 31, 2003 @11:13PM (#6087849)
    Fuck! <<shreds notebooks full of groundbreaking work which assumes the proof was good>> Back to my job at the gas station, I guess....

This place just isn't big enough for all of us. We've got to find a way off this planet.

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