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Another Breakthrough in Prime Number Theory
Battal Boy writes "From aimath.org: Dan Goldston and his Turkish colleague Yalcin Cem Yildirim have smashed all previous records on the size of small gaps between prime numbers. This work is a major step toward the centuries-old problem of showing that there are infinitely many 'twin primes': prime numbers which differ by 2, such as 11 and 13, 17 and 19, 29 and 31,...I am especially proud of this achievement as Yalcin is a close friend of mine from way back! You may also want to check out the Mercury News Article and Dan Goldston's home page where you can see a photo of Dan's back being slowly but surely broken by two of his children ..." Finding patterns in primes seems to be all the rage.
Zzzzzzzzzzzzzzzzzzzzzzzzz.
;-)
How about calculating the rate of ring growth in trees as well? How about the speed that paint will dry at various temperatures?
Ok, so there are infinite twin prime numbers but what about secondary numbers? Have we just given up on them?
Checking out my form of escapism.
Here I thought the patterns-in-primes thing was already solved by Jodie Foster and Matthew McConaughey..
This is certainly a signficant advance in mathematics... prime numbers are one of the most enigmatic, yet useful aspects of number theory. What I'm really curious to see is whether or not this will help the efforts to find a more efficient algorithm for factoring a number into its prime factors. (A multiple of two very large primes is an integral part of RSA encryption, as well as other schemes.)
-- Apple: Where Microsoft wants to go today.
Funny, I can't see their server getting slashdotted anytime soon...
The scientific community was shocked today to learn that there exist not two but THREE prime numbers between 4 and 8.
sentient robot will probably die laughing when it reads this research
Do not try to read the dupe, thats impossible. Instead, only try to realize the truth
What truth?
There is no dupe
Are there any?
Lesbian Nazi Hookers Abducted by UFOs and Forced Into Weight Loss Programs - -all next week on Town Talk.
Patterns! Maybe they should get Maximillian Cohen involved!
=Smidge=
There's a small joke that goes around in the academic world
Biologists like to think they are chemists. Chemists like to think they are physicists. Physicists like to think they are mathematicians. And mathematicians like to think they are god.
Seriously, think of how much of what we learn boils down to our understanding of numbers, systems, and patterns within them. Mathematics, whether you like it or not, is really a beautiful and elegant subject that very few truly understand.
2^3-1 is prime! Now checking 2^3+1...
150 Opening BINARY mode data connection for slashdot.sig (129323052 bytes).
I was waiting for a better time to break this, but I guess now is good to. I have made a groundbreaking discovery in prime numbers.
No prime numbers can be divided by any number that falls inbetween the number one and the number itself! And, even more exciting, a rule that applies to all prime numbers. All prime numbers can be factored with the number one, but none can be divided by zero.
I hope none of you had anything important "encrypted" with PGP. Just stick to padless one-time pads for *real* security.
After I get the National Math Foundation to classify two as an odd number (and it is really odd considering it's the only even prime number) I'll have my third discovery that all prime numbers are odd validated.
I'd love to post more, but I really must get back to working on my perpetual motion machine. I was so close before, but recently I seem to have lost my bearings. Once I'm done I'll be heralded as the greatest man in the realm of science friction.
Have you read my journal today?
NEWSFLASH! There are infinitely many numbers in the universe! Current research held thaat numbers only went up to a few hundred trillion, but scientists have recently discovered that numbers continue FOREVER! ...
here
Go on, how many ways of measuring the height of a building using a barometer can /. come up with?
This is more interesting than watching paint dry actually! I am waiting for the the paint to dry in my kitchen and i'm reading this instead :-)
Its one damn thing before another. (Dick Bird 1999)
This has nothing to do with encryption, nothing to do with RSA, nothing to do with practical applications at all. Factoring and cryptography is only a small part of the ocean that is number theory. Please don't automatically assume that anyting about number theory or prime is related to encryption and practical applications. This one certainly isn't. This is about twin primes: the authors have proven that the gaps between consecutive primes are small, asymptotically smaller than the logarithm of the number. This *might* lead to attacks on the twin prime conjecture. Nothing is known yet. This is highly theoretical work. Appreciate pure mathematics, with its beauty, for its own sake.
Do negatives count? Or do 3 and -3 count as factors? And if this is so, then what stops a positive number from being composite when its factors can include its negative and -1?
...have smashed all previous records on the size of small gaps between prime numbers
The size of small gaps? For a start they couldn't really "smash" the size of small gaps because then by definition they would no longer be small ones. But secondly how exactly can you smash all previous records on the size of the small gaps between the prime numbers when it is set to (and the whole theory based around) the gap being "2".
I'm not an expert by any means, but surely if what the story means is they have found more of these pairs (1000928397 and 1000928399 or something -- and no, I just made them up)... then, so what? Sure it's an achievement of sorts, but it doesn't really help? To really help then a proof that this does or doesn't go on to infinity is what is needed.
... I guess
> This work is a major step toward the
> centuries-old problem of showing that there are
> infinitely many 'twin primes':
If the goal is indeed to prove that there are an infinite number of twin primes demonstrating the existence of any finite number of them, however large, is no step at all.
Warning: this article may contain humor, sarcasm, parody, and perhaps even irony. Read at your own risk.
One reason that it is intuitivly possible that there are an infinite number of twin primes is that it is possible to generate numbers that are relativily prime. For example, multiply the first three primes: 2,3, and 5. You get 30. Add or subtract one from 30 and you will get a number that is relatively primet to 2, 3, and 5. In this case you get 29 and 31. Both happen to be prime numbers. We've found a twin.
The hard part of proving there are an infinite number of twins is finding a way of showing that your relatively prime numbers are truely prime. IE, in this case that neither is divisible by a higher prime such as 7, or 11.
Next step is to find prime numbers differing by 1.
Can anyone explain what either of these things signifies or what their impact is...for non-mathematicians?
There is an infinite number of twin primes. Can we go blame Microsoft now?
How exactly does one hold on to frictionless bearings? Do you use [http://www.archive.org/movies/details-db.php?id=2 74]Johnson & Johnson plastic wraps to stick to them?
Several scientists were asked to prove that all odd integers higher than 2 are prime.
..."
http://www.math.utah.edu/~cherk/mathjokes.html
Mathematician: 3 is a prime, 5 is a prime, 7 is a prime, and by induction - every odd integer higher than 2 is a prime.
Physicist: 3 is a prime, 5 is a prime, 7 is a prime, 9 is an experimental error, 11 is a prime. Just to be sure, try several randomly chosen numbers: 17 is a prime, 23 is a prime...
Engineer: 3 is a prime, 5 is a prime, 7 is a prime, 9 is an approximation to a prime, 11 is a prime,...
Programmer (reading the output on the screen): 3 is a prime, 3 is a prime, 3 a is prime, 3 is a prime....
Biologist: 3 is a prime, 5 is a prime, 7 is a prime, 9 -- results have not arrived yet,...
Psychologist: 3 is a prime, 5 is a prime, 7 is a prime, 9 is a prime but tries to suppress it,...
Chemist (or Dan Quayle): What's a prime?
Politician: "Some numbers are prime.. but the goal is to create a kinder, gentler society where all numbers are prime... "
Programmer: "Wait a minute, I think I have an algorithm from Knuth on finding prime numbers... just a little bit longer, I've found the last bug... no, that's not it... ya know, I think there may be a compiler bug here - oh, did you want IEEE-998.0334 rounding or not? - was that in the spec? - hold on, I've almost got it - I was up all night working on this program, ya know... now if management would just get me that new workstation tha just came out, I'd be done by now... etc., etc.
Saskboy's blog is good. 9 out of 10 dentists agree.
Is it supposed to be "for Pete's sake" (i.e. for the sake of Pete who is stupid) or is "petesake" a real expression?
Given that prime numbers are infinite, and given that the regularity with which they occur does not seem to have any set frequency, can we not say twin primes are also (likely) infinite?
Obviously that is not at all rigorous but it strikes me as being true.
--Joey
Indeed, take Buffon's Needle Problem for instance. whoda thunk it.
"Not knowing when the dawn will come, I open every door." - Emily Dickinson
No way! The odds of such a discovery are about the same as running Linux on an XBox without a modchip! It just can't happen!
How exactly does one hold on to frictionless bearings?
Actually, after discovering the real answer of the square-root of two, I filtered some dirt, dehydrated some water, and desalinated some salt. I then came up with *exactly* what would hold them.
Have you read my journal today?
Drop it and time it's fall. Be sure to account for the speed of sound or light depending on how you detect it breaking.
You can also go to the owner of the building and say, "I'll give you this barometer if you tell me how tall your building is."
I second that!
You want boring? Then go and take a look at the PDF papers on this site.
Are they boring? Yes, exruciatingly and mind numbingly so...
Did they help us win the Second World War? err...yes
A cynic is what an idealist calls a realist...
Does it run Linux?
Will ths in any way help me get out of The Cube?
All work and no play makes me a dull boy
...who cares.
"I'll give you this barometer if you tell me how tall your building is."
Someone has been reading Strata...
Everything in the world is controlled by a small, evil group to which, unfortunately, no one you know belongs.
Mathematicians described the advance -- announced at a conference in Germany -- as the most important breakthrough in the field in decades.
Personally, I think the technique for provably determining the primality of an arbitrary number in polynomial time -- "PRIMES is in P" -- was a more unexpected result. It's always seemed like the probability of a twin prime occuring on the number continuum was a limit approaching but never quite reaching 0 -- an artifact of the number of previous primes already "exposed" approaching, but never reaching infinity. But actually sitting down and proving this -- excellent! Very cool.
--Dan
I believe this is a link to the paper...0 /2000-014/2000-014abs.html
ftp://ftp.msri.org/pub/publications/preprints/200
Not too sure but looks like a preprint of the thing
.ACMD setaloiv siht gnidaeR
Some activities give you immediate emotional feedback.
More importantly, he should make sure NOT to shoot down the stalactite in the Gravity Chamber. There's a missile expansion in there, and shooting down the stalactite exposes a grapple point to get to the missiles.
However, if you shoot that down, and leave the room without claiming the missile expansion, then the grapple point is dead forever after. Since you can shoot that thing down long before you get the grapple beam, this is easy to do.
If you do this, you can still beat the game, you just can't reach 100% items and get the third ending. That really pissed me off when I reached 99%.
I surfed away from Mothers Against Boomerangs for this?
Seriously, I didn't care about primes when I had to, I really can't see a compelling reason to look for really big numbers that don't factor except for sheer curiosity. To me this is like trainspotting for dollars.
Wake me when there's an application.
Don't wake me for troll modding. I know it's coming.
And erm, proving that there's an infinite number of them? How will you know when you're done?
And hunting for primes? I'm calling People for the Ethical Treatment of Numbers.
"Win treats sysadmins better than users. Mac treats users better than sysadmins. Linux treats everyone like sysadmins."
From the Mercury News:
``Mathematicians described the advance -- announced at a conference in Germany -- as the most important breakthrough in the field in decades."
Oh, & proving Fermat's Last Theorem in 1995 was just another undergraduate exercise?
(For the non-math-nerds, proving true Fermat's Theorem -- that the formula a^n * b^n = c^n was insolveable where n is greater than 2 -- was considered for three _centuries_ to be _the_ principal challenge in mathematics. The man who did this -- Andrew Wiles -- spent about 30 years working on this, & succeeded only after a second try.)
Geoff
I think I see a trend here. Maybe for them it really would be easier to muzzle the entire internet than to produce p
Small gaps between consecutive primes
:= lim_inf(n -> +inf, (p_(n+1) - p_n) / log(p_n))) = 1
Recent work of D. Goldston and C. Yildirim
What are the shortest intervals between consecutive prime numbers? The twin prime conjecture, which asserts that p_(n+1) - p_n = 2 infinitely often is one of the oldest problems; it is difficult to trace its origins.
In the 1960's and 1970's sieve methods developed to the point where the great Chinese mathematician Chen was able to prove that for infinitely many primes p the number p + 2 is either prime or a product of two primes. However the well-known ``parity problem'' in sieve theory prevents further progress.
What can actually be proven about small gaps between consecutive primes? A restatement of the prime number theorem is that the average size of p_(n+1) - p_n is log(p_n) where p_n denotes the nth prime. A consequence is that
delta
In 1926, Hardy and Littlewood, using their ``circle method'' proved that the Generalized Riemann Hypothesis (that neither the Riemann zeta-function nor any Dirichlet L-function has a zero with real part larger than 1/2) implies that delta = 2/3. Rankin improved this (still assuming GRH) to delta = 3/5. In 1940 Erdös, using sieve methods, gave the first unconditional proof that delta 1. In 1966 Bombieri and Davenport, using the newly developed theory of the large sieve (in the form of the Bombieri - Vinogradov theorem) in conjunction with the Hardy - Littlewood approach, proved delta = 1/2 unconditionally, and then using the Erdös method they obtained delta = (2 + sqrt(3))/ 8 = 0.46650... . In 1977, Huxley combined the Erdös method and the Hardy - Littlewood, Bombieri - Davenport method to obtain delta = 0.44254. Then, in 1986, Maier used his discovery that certain intervals contain a factor exp(gamma) of more primes than average intervals. Working in these intervals and combining all of the above methods, he proved that delta = 0.2486, which was the best result until now.
Dan Goldston and Cem Yildirim have a manuscript which advances the theory of small gaps between primes by a quantum leap. First of all, they show that delta = 0. Moreover, they can prove that for infinitely many the inequality
p_(n+1) - p_n (log(p_n))^(8/9)
holds.
Goldston's and Yildirim's approach begins with the methods of Hardy-Littlewood and Bombieri - Davenport. They have discovered an extraordinary way to approximate, on average, sums over prime k-tuples. We believe, after work of Gallagher using the Hardy-Littlewood conjectures for the distribution of prime k-tuples, that the prime numbers in a short interval [N, N + lambda log(N)] are distributed like a Poisson random variable with parameter lambda. Goldston and Yildirim exploit this model in choosing approximations. They ultimately use the theory of orthogonal polynomials to express the optimal approximation in terms of the classical Laguerre polynomials. Hardy and Littlewood could have proven this theorem under the assumption of the Generalized Riemann Hypothesis; the Bombieri - Vinogradov theorem allows for the unconditional treatment.
This new approach opens the door for much further work. It is clear from the manuscript that the savings of an exponent of 1/9 in the power of log(p_n) is not the best that the method will allow. There are (at least) two possible refinements. One is in the examination of lower order terms that arise in his method. Can they be used to enhance the argument? The other is in the error term Gallagher found in summing the ``singular series'' arising from the Hardy-Littlewood k-tuple conjecture. There is reason to believe that this error term can be improved, possibly using ideas in recent work of Montgomery and Soundararajan (``Beyond Pair - Correlation''.)
It is not clear just how far this method can be pushed and what other problems might be attacked using his new ideas; at this point we can't rule ou
Yea, linux is great, run factor on common numbers. If I remove the last digit from my SSAN, it's a prime number! Add the last digit and it's a fizzle. SO! All you perl, sed, gawk, gasp, list, C, C++ hax0rs get busy, I need something that takes a number and runs many permutations on it factoring each! Removing an the ending digits, one by one and factoring them. The beginning digits... Hey, we could have our on Bibble code! Stir in a laddling of The code of Hammurabi, KJV Gute, K, cOs, rederered into mumbers of course and we will answer all, I mean all of the questions.
I'd go on a Vegan diet but the delivery time from Vega is too long. --brownkitty
The topic character limit prevented that. or so it seemed.
If they were all tied together, would this mean that the structure of living creatures could be explained with mathematics? (in theory of course)
The day this guy dies, he will regret wasting his time on this
Partners in Prime ;)
Large prime numbers tend to form Beowulf clusters.
There's a small joke that goes around in the academic world
Biologists like to think they are chemists. Chemists like to think they are physicists. Physicists like to think they are mathematicians. And mathematicians like to think they are god.
FYI, there's another version of that joke which ends on "philosophers".
Boring? (Score:3, Interesting)
Congrats. You have the first oxymoronic post title on Slashdot.
Having your score as the second prime was just icing on the cake.
Soko
"Depression is merely anger without enthusiasm." - Anonymous
Quoting from the source code of the applet:
Anybody still amazed that the result of the algorithm is pi?
Reminds me of the old strings -a /proc/kcore | fgrep "weapons of mass destruction" joke, let's call it Bush's grep command.
I think you meant to say that "Engineers are GOD!", that is to say that we are the closest thing since no god actually exists.
----
Go canucks, habs, and sens!
I'm not a mathematician, but mathematics will always be my first love (which might explain the lack of success in <ahem> another area...) :-)
Everyone's talking about the practical value or lack thereof of this discovery wrt crypto, but can't we just appreciate this for the beauty of its contribution to number theory? I'm no mathematician, but new disoveries in number theory rank higher on the interesting scale to me than the average /. article...How is this in any way 'boring'?
They were once only divisible by 1 and P. Now it turns out they're all divisible by -1 and -P as well. Dammit, all the textbooks are going to have to be changed!
-- thinkyhead software and media
philosophers think they are god and then they claim "I don't exist".
The best education consists in immunizing people against systematic attempts at education. - Paul Feyerabend
You know you're in over your head when you don't understand why comments modded "funny" are funny.
It can't be a coincidence that his kids' ages are all twin primes. Can it?
Never attribute to stupidity what can be construed as a monopoly preservation tactic.
Prime numbers are defined to be both integers and positive.
At least, that's what Eric Weisstein thinks.
So do The Prime Pages
Gotta be positive, por favor.
Support a few technologists in Washington.
It is interesting to note that the ages of his three kids are all consective primes:
3, 5 and 7.
Research is what I doing when I don't know what I am doing - Werner von Braun
I just want to know whether there are any odd perfect numbers.
god dammit. All the stupid less signs are gone. I didn't notice it when I posted. Ok. Basically, everywhere where it says "delta = ..." it should read "delta <= ...". The inequality should read p_(n+1) - p_n *less than* ... and so forth.
I think you're grossly overestimating how many people truly understand mathematics. I'll venture 'no one' as the true answer to that, and ask for any counter examples.
"Who is the Journal of Quantum Physics going to believe?" --Stephen Hawking
0 does not divide anything, so the implication
0|(a*b) => 0|a or 0|b
is vacuously true (even if you consider that 0|0 is true, the implication still holds since a*b=0 => a=0 or b=0).
Similarly, for p = 1:
1|(a*b)
=> a*b != 0
=> a != 0 or b != 0
=> 1|a or 1|b
The same argument for the development of the atom bomb holds true for crypgology. If "super-smart but real-world-naive" researchers don't do the research and publish it, the NSA, DARPA or some similarly real-world-non-naive researcher is going to do the research, and the results will be classified.
Somebody will always be looking for a way to gain power over somebody else. This somebody can be "the good guys", "the bad guys", or whatever term you want to use.
...but 31 and 41 differ by 10, not 1.
And you understand mathematics well enough to judge that others do or don't understand mathematics?
t
Is it supposed to be "for Pete's sake" (i.e. for the sake of Pete who is stupid) or is "petesake" a real expression?
im pretty sure its "Pete's sake". There was a movie named that too. Barbara S. invests in pork bellies. Thats all I remember of it since I don't like her.
Tequila: It's not just for breakfast anymore!
> Do you think he had sex with a woman during that entire period?
r ew _wiles.htm
I don't know - you might want to ask his *children*:
http://mathematics.gc.cc.fl.us/mathprojects/and
[Sarcasm] I want to start a process where I prove that all even numbers are, in fact 'even' or divisible by two. I'll need a worldwide network of computers to set about to prove this....all the way to the last number! [/Sarcasm] Balls. Scan for anomalous radio signals. Study the growth patterns of wheat over the last 100 years, use it to win the lottery. There's not much less useful that that old saw, factoring prime numbers, sorting prime numbers, using prime numbers in wallpaper patterns, or brushing one's teeth with prime numbers. See also 'Chaos theory': Measure some activity transacted with infinite precision, measure it in only 8-digit precision, and then look like an ape when you can't predict what's going to happen...."it's chaos".... C'mon, guys- start using that intellect and curiosity for real-world stuff.
--- For a good time mail uce@ftc.gov
And God created man (and all that other biology stuff)
The number 2 is prime, and is indeed even. Don't quit your day job.
...Also, I didn't know Buggalo could fly.
I only skimmed your post before replying, didn't notice the qualifier you had. Sorry!
...Also, I didn't know Buggalo could fly.
I love mathematics (especially stuff like Category Theory, Lambda-calculi, etc...), but why does Slashdot always post number theory stories? You can try to view all of math through numbers, but you are missing out on allot of structure by doing so.
There are also no prime number triples besides the first one: 3,5,7
I thought it was biologists like to think they are chemists, chemists like to think they are physicists, physicists like to think they are god, and god likes to think he's a mathematician.
You take all the primes currently and multiply them together to get x. X+1 and x-1 are bot prime and therefore twinprimes. I know x+1 is prime but why wouldn't x-1 also not be prime?
Why don't you guys have friends or journals?
Biologists like to think they are chemists. Chemists like to think they are physicists. Physicists like to think they are mathematicians. And mathematicians like to think they are god. /.
and programmers like to think they are engineers as seen today on
...as well as people discussing (maybe even arguing) over whether the number "1" is or is not a prime.
Reason is the Path to God - Anon
It's been a while since I studied RSA prime selection, but I'm sure someone will rush to correct my errors.... :-)
With RSA, I thought you wanted "strong primes," not just primes. A strong prime p is one such that p = 2p' + 1, where p' is also prime. This means that Phi(pq) = (p-1)(q-1) = 4p'q'.
Anyway, in practice this means that you'll go through a lot of primes before you find one suitable for use in an RSA key. That's why it takes so long to generate an RSA keypair....
For every complex problem there is an answer that is clear, simple, and wrong. -- H L Mencken
If you multiply all primes up to any prime, and the add or subtract 1, both resulting numbers are prime. Since there are infinitely many primes, there must be an infinite number of twins.
For instance, (2*3)-1 = 5 and (2*3)+1 = 7.
(2*3*5)-1 = 29
(2*3*5)+1 = 31
(2*3*5*7)-1 = 209
(2*3*5*7)+1 = 211
--Justin Mitchell
"2nd Place is a fancy word for losing" --Bender (Futurama)
The regular Ice Beam was fine... just make sure to get Prime off of Ice mode as quickly as possible... I never even found the Ice Spreader, and I only died once before beating it.
This actually has incredable reprucussions on the world of computer AI. Many AI engines these days make great use of prime numbers, as well as twin prime members to develop decision trees. As an AI developer, my work would be impossible without the knowledge of twin prime numbers. Most software you have ran into these days with AI depends upon the use of twin prime numbers.
This is an incredable development to the AI community, as we now know for sure that we have an infinate number of twin prime numbers.
"Entropy is the bad-guy, and he is everywhere"
There are also no prime number triples besides the first one: 3,5,7 Okay. Where's your proof?
Who moderates the meta-moderators?
http://www.bearnol.pwp.blueyonder.co.uk/Math/index .html
I read it in the paper a while ago and was surprised to see anything of that magnitude come out of my school...especially from the building known to the students as the "Insane Asylum" (quite a few math instructors and professors have....well..."issues." Hell all of the calc professors had all unique "issues" [one, who was manic-depressive, was cool])
Oh well....I'm sure this will help future algorithms.
"A hint of his sense of humor can be found on his Web site, which features a photo of Goldston, seemingly dozing off, as two small kids climb on his back. He and his wife, Ryoko, have three children -- Shota, 7, Aiko, 5, and Makoto, 3."
Can we can expect his next theorem to deal with prime triplets?
2*3*5*7 - 1 is NOT prime.
2*3*5*7*11*13 + 1 is NOT prime.
2*3*5*7*11*13*17 +/- 1 are BOTH not prime.
It is highly uncommon for these primorials (as they are called) to be a prime plus or minus 1. There are only a few dozen cases known up to 100000.
I'm giving the direct link.t ml
http://www.cse.iitk.ac.in/news/primality.h
Every triple of that form contains at least one number that can be divided by 3. If you want all numbers to be prime, the number 3 itself has to be among the three numbers, hence the triple 3,5,7, is the only one. (short idea of the proof, the full one would require a few more lines of text)
I intend to live forever, so far so good.
So what about the gap between the two prime numbers 1 and 2? It would seem that there is not only one pair of primes with such a small gap.
Remind me not to get you to do my encryption.
Engineer: 3 is prime, 5 is prime, 7 is prime, 9 is prime, 11 is prime, 13 is prime, 15 is prime.....
Yes. I understand mathematics well enough to be confident that no one has but the tiniest fraction of a grasp on the whole. There are a tiny, tiny number of people (probably less than a million worldwide) who know math perhaps one thousand times better than I do at most, and even that doesn't begin to touch true understanding. It's a 'the more you know, the more you realize you don't know' situation. Seriously, ask any mathematics PHD if they feel like they are even aware (much less understand) half of what has been proven in mathematics.
"Who is the Journal of Quantum Physics going to believe?" --Stephen Hawking
As much as I'd love to have come up with something novel, this really isn't.
a *b)9 99999 99
:-)
Take the limit as p approaches infinity of sqrt((p-1)(p+1)). It basically reduces to sqrt(p*p) = sqrt(p^2) = p.
So as p reaches infinity, the impact of those +- 1's goes to 0. The limit is p. p doesn't even need to be prime:
a=9845769847569845645692837498235
b=a+2
sqrt(
9845769847569845645692837498235.999999999999
You of course need all the other tricks for non-twin primes
--Dan
once quantum computer appear that will be the death of rsa. rsa is dead long live quantum encryption
Biologists like to think they are chemists. Chemists like to think they are physicists. Physicists like to think they are mathematicians. And mathematicians like to think they are god.
And god likes to think he's a biologist.
or
Computer scientists like to think they killed god.
"Probably the toughest time in anyone's life is when you have to murder a loved one because they're the devil." -Philips
Worse, there are those who believe that if there is difference between the mathematical model and experimental data, the experiment is wrong.
Now, there are cases where an experiment is flawed, or, in the case of measuring things like the mass of the neutrino, the value is so small, experiments have grand difficulty getting any accuracy, but, overall, REALITY is real, not someones mathematics. Mathematics is a model, a contrived one at that. No real surprise that there are elegant things to be found.
I have this argument with a physicist friend of mine about once a year. He always presumes the math is right. He is usually proven wrong months later by yet another discovery.
Sigh.
Shannon Mann.
A comment overheard in a corn field `If you have better ideas, lets hear them. I am all ears.'