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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.
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!
Scott
I have no idea how this proof works because the server melted already.
Wh47 d1d j00 541, 31337 15n't t3h r0xor5 ne m0r3???
Twin primes are two prime numbers that differ by a value of two - for instance, 17 and 19, or 29 and 31.
but the space that I'm allowed to type in here is too short.. :-)
There's a gorilla from Manilla whose a fella that stinks of vanilla and has salmonella.
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
/bin/fortune | slashdotsig.sh
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.
Yeah, right, like I can imagine CmdrTaco rejecting a story because he read the math and found the error.
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!
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.
The correlation between ignorance of statistics and using "correlation is not causation" as an argument is close to 1.
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???
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.
Mod down posts with a "Free Mac Mini/iPod" sig, they're spam!
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.
Note to M1-ers: a curt but otherwise insightful message is not "Flamebait" or "Troll".
27159925611: 3 7 13 13 17 19 19 29 43
$ echo '13*17*19*29*57*91*43' | bc
27159925611
Thus, on the command line, the factorization is easier!
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.
make world, not war
Your example is pretty poor, in the sense that special-purpose factoring algorithms (Pollard rho, Pollard p-1, Lenstra's ECM) can comfortably factor numbers with many small factors, regardless of the number's size. In fact, ECMing 40-digit prime factors out of numbers with tens of thousands of digits is commonplace today, as is ECMing smaller-sized factors from numbers of millions of digits.
Now factoring a number 200 digits long with only two (and equally-sized) factors would be a world record.
Join the NFSNET. Our prime goal is making little numbers out of big ones. http://www.nfsnet.org/
2+2 does equal 5, for sufficiently large values of 2.
:)
I love being a mth dork
Matt Fahrenbacher
James Tiberius Kirk: "Spock, the women on your planet are logical. No other planet in the galaxy can make that claim."
What would be really important is to prove the Reimann Hypothesis. That would tell us a lot about the distribution of primes.
Yeah, and it's not even close. "order of magnitude" is more popular by a... heck of a lot. (heh.)
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).
This post is dedicated to all of those
It was in here.
Unfortunately, I devoured it. Damn you Bill Cosby!
No sweat: 4294967279 * 4294967291
Everybody knows, that the best tool for factoring numbers is google:
http://www.google.ca/search?q=18446743979220271189
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.
Unlimited growth == Cancer.
Fuck! <> Back to my job at the gas station, I guess....
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 Theorem: "if and only if p is a prime, then (p-1)! +1 is a multiple of p, that is (p-1)! congruent to -1 (mod p)." (quotations from mathworld.com) Theorems such as these provide insight into the distribution of primes throughout the natural numbers. The Twin Prime Conjecture, if resolved, would provide additional insight into this distribution, which would be of fundamental theoretical and practical importance. For instance, it is currently regarded as hopelessly time consuming to factor large composites--public key cryptography is based on this fact. But I am not aware of any proof that factoring such numbers *must* take a long time--that they do is an interesting state of affairs, but it might not reflect the nature of the universe so much as our lack of knowledge about prime numbers. Solving the TPC would be a step in remediating that deficiency.
"Oh, the tragedy of math gone wrong. I can't even talk about it." -Wil Wheaton http://www.wilwheaton.net