Slashdot Mirror
Twin Prime Proof Erroneous
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."
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.
Twin primes are two prime numbers that differ by a value of two - for instance, 17 and 19, or 29 and 31.
aimath.org/primegaps/
aimath.org/primegaps/residueerror/
I'm still working on mirroring all 47 images, but the text is there, and the img tags have great alt text descriptions.
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.
Wh47 d1d j00 541, 31337 15n't t3h r0xor5 ne m0r3???
No sweat: 4294967279 * 4294967291
Everybody knows, that the best tool for factoring numbers is google:
http://www.google.ca/search?q=18446743979220271189
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
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, 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, 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 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 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: