Twin Prime Proof Erroneous 199
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."
Idea may lead to new record, not twin prime proof (Score:5, Informative)
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.
Re:anyone got some asprin? (Score:5, Informative)
Re:reminds me of a bad math joke (Score:3, Informative)
Because he left his residue at every pole
Ducks
mirror (Score:5, Informative)
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.
Re:A serious question - i'm not trolling, honest! (Score:5, Informative)
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.
Re:A serious question - i'm not trolling, honest! (Score:5, Informative)
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.
Re : Necessity of calculating pi? (Score:5, Informative)
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.
Re:A serious question - i'm not trolling, honest! (Score:4, Informative)
Now factoring a number 200 digits long with only two (and equally-sized) factors would be a world record.
Re:A serious question - i'm not trolling, honest! (Score:3, Informative)
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
Re:The sibling comments are being wiseass. (Score:5, Informative)
No sweat: 4294967279 * 4294967291
Everybody knows, that the best tool for factoring numbers is google:
http://www.google.ca/search?q=18446743979220271189 [google.ca]
Re:Maths jokes = Instant karma! (Score:5, Informative)
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:
Re:Maths jokes = Instant karma! (Score:4, Informative)
The joke is referring to the Laplace transform. There is no Lacrosse transform.
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.
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.
Re:A serious question - i'm not trolling, honest! (Score:3, Informative)