Major Advance Towards a Proof of the Twin Prime Conjecture 248
ananyo writes "Researchers hoping to get '2' as the answer for a long-sought proof involving pairs of prime numbers are celebrating the fact that a mathematician has wrestled the value down from infinity to 70 million. That goal is the proof to a conjecture concerning prime numbers. Primes abound among smaller numbers, but they become less and less frequent as one goes towards larger numbers. But exceptions exist: the 'twin primes,' which are pairs of prime numbers that differ in value by 2. The twin prime conjecture says that there is an infinite number of such twin pairs. Some attribute the conjecture to the Greek mathematician Euclid of Alexandria, which would make it one of the oldest open problems in mathematics. The new result, from Yitang Zhang of the University of New Hampshire in Durham, finds that there are infinitely many pairs of primes that are less than 70 million units apart. He presented his research on 13 May to an audience of a few dozen at Harvard University in Cambridge, Massachusetts. Although 70 million seems like a very large number, the existence of any finite bound, no matter how large, means that that the gaps between consecutive numbers don't keep growing forever."
Preprints not avaiable, but it seems legitimate (Score:5, Informative)
The paper seems to have been accepted by Annals of Mathematics, which is basically the number one mathematics journal.
Also, according to New Scientist, Henryk Iwaniec (a well-known analytic number theorist) has reviewed the paper and didn't find an error. This may or may not overlap with the review at Annals, though.
Newbie question (Score:2)
I have a question: (excuse me for the realy bad formatting)
If the result of prime numbers (plotted), can be formulated as e^x, where Xaxis = numbers (zero to infinity) v Yaxis = [amount of unique distances observed] ; ; ;
and plotted against the plotting of prime numbers themselves
and plot a formula_3 that averages the coordinates that both euclidean functions output, towards infinity, through where they almost intersect
and form a formula_4 that equals the offset of the two euclidean function, relative to f
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"Doesn't it make a lot of sense?"
No.
Not in North Carolina (Score:5, Funny)
No siree. Ain't non prime numbers at all here in North Carolina since we done banned them. Ain't no angels felled out of the sky, ain't no computers breakin', and my cousin's kisses never tasted sweeter. Prime numbers are a godless socialist conspiracy against Jedus and mah wallet.
Re:Not in North Carolina (Score:5, Funny)
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So I guess in North Carolinian schools they only teach twin primes that consist of one odd prime and one even prime.
Stories like this... (Score:2, Interesting)
Stories like this only remind me of how ignorant I still am and how I've wasted my life.
Re:Stories like this... (Score:5, Funny)
Stories like this only remind me of how ignorant I still am and how I've wasted my life.
Don't feel bad. Maybe you've made coffee for, served fries to, or unclogged the toilet of one of these great people? Every little bit helps!
Re:Stories like this... (Score:5, Insightful)
Re:Stories like this... (Score:5, Insightful)
Re:Stories like this... (Score:5, Insightful)
It's shoulders all the way down.
You were probably going for funny, but if I had mod points I'd call this insightful. It really is shoulders all the way down; no one accomplishes anything of significance without relying on many, many others.
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Part of the art of humor is in being insightful while being funny. It why the greats are so great, they just tell you good life advice, but they frame it in an amusing way and we love them for it.
Re:Stories like this... (Score:5, Insightful)
Um, one question that a person could ask is: If this proof is found, how does it change the world? How would being able to use the proof influence something in the real world? I'm not saying it can't or won't, only that simply picking a brainy subject does not mean that doing things in it aren't basically intellectual masturbation.
The change to our world is this: we now know something that we didn't know before. Now we can teach this new knowledge to others (and by others I mean people smarter than me) who can find new places and ways to apply this new knowledge. They might never do anything interesting with it, or it might cause an avalanche of new findings, we don't know. But we, as a species, fundamentally know more today than we did yesterday.
As an example, the ancient greeks studied prime numbers. Was there any immediate use of primes at the time? Did it allow them to improve harvest? Defeat the Roman army? Nope, they just studied them. At the time there is no way that they could have conceived their application for encryption. Yet today, all commerce on the web uses the mathematics of primes.
It is not important to have an immediate use for knowledge.
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If this proof is found, how does it change the world?
Some encryption algorithms are based on large prime numbers. But how to you find a large prime number? One method often used is to pick a random large odd number, test if for primality, and if it is composite, increment by two and try again. The problem with this linear search method is that you are far more likely to pick a prime that is separated by a large gap from the previous prime, and you will almost never pick the larger prime of a twin pair. So a better understanding of the gaps between primes
Gaps between numbers... (Score:5, Funny)
To be perfectly honest the proof that the gap between consecutive integers doesn't grow forever is pretty simple. It stays 1.
Re:Gaps between numbers... (Score:5, Informative)
Joking aside, submitter is not a mathematician. This doesn't prove anything about the gap between arbitrary consecutive primes. That gap does indeed grow forever, by the known distribution of primes, but by "chance" one would expect a few pairs to lie close together. The proof is that this "chance" event still occurs as N tends to infinity. The same result would hold for random numbers whose distribution gets more sparse with increasing N so it just says that the primes are not "less random" than these (in a very informal sense).
Re:Gaps between numbers... (Score:4, Insightful)
The same result would hold for random numbers whose distribution gets more sparse with increasing N
This is false --- depending on how fast the random numbers "spread apart", you can have an infinite number of random numbers but a finite number of "close pairs". Simple example: for each positive integer N, choose N to be in your set of random numbers with probability 1/N. This gives you an infinite expected number of such random choices: sum 1/N over positive integers diverges. But what's the chance of adjacent pairs? The probability of N and N+1 being in your random set is 1/N * 1/(N+1). The expectation value for this set is *finite*: sum 1/(N*(N+1)) converges to a finite value.
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Given the bigger the number, the more smaller numbers that could divide into it, suggests primes should be getting further and further apart.
Further apart _on average_, which they do. This result doesn't change that; the density of primes still goes to zero.
The question is whether despite that increasing _average_ distance, there is also an increase in the _minimum_ distance. That was unproven, and now we know: the greatest minimum distance is at most 70 million. It could be as small as 2, and in fact I think most mathematicians strongly suspect that it is: no matter how high you go, there will always be some pair of primes larger than that, sep
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Yes. And I was joking, not serious. Duh!
Re:Gaps between numbers... (Score:5, Funny)
Re:Gaps between numbers... (Score:5, Funny)
He may be English.
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Traditionally, it helps when the recipient has a sense of humor, too.
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Let me demonstrate.
"You want me to fix your toaster? Hold it firmly and run hot water over it, while putting its plug in the power outlet."
This is only funny if you get it.
Humor is suggesting a (absurd yet understandable) relation between unlikely things.
It also needs a degree of originality and unexpectedness to it - absurdity is not enough. I found rew's comment funny because I wasn't expecting that reply.
The toaster thing isn't funny at all. It's just nasty. Water + electrical appliances have been way overdone in both pop culture and humour.
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Its just what that US science guy once said, to highlight differences in understanding simply due to insights.
And yes, originality/unexpectedness is key, i rather unsuccessfully implied that with "absurd yet understandable".
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Given that the only "real use" for large primes is cryptography, I was in my nerd mindset thinking that this means that there will now be a near finite amount of processing power required to break algorithms.
However, I keep comming back to xkcd aswell: http://xkcd.com/538/ [xkcd.com]
Conclusion wrong (Score:4, Informative)
"the existence of any finite bound, no matter how large, means that that the gaps between consecutive numbers don't keep growing forever"
Actually, I disagree with the unfortunate writing of the sentence. The gaps between consecutive prime numbers are variable, and on average they DO tend to keep growing forever. This is a widely known result, the density of prime numbers decreases as the numbers grow. However, since the gap between consecutive primes is variable and it does not follow a regular function (otherwise, it would be very easy to calculate prime numbers), even with a very low density of prime numbers we can find a pair of consecutive prime numbers with a gap of only 2.
The problem under study is not wether the gap between consecutive primes keeps growing forever (which is true only on average, considering a long secuence of integers), but wether there are infinite such pairs of primes with gap 2. The new result found says that there exist infinite pairs of primes with gap 70M or less. However, this does not imply at all that no consecutive pairs of primes with gap > 70M exist (which, in fact, they do).
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It /is/ very easy to calculate prime numbers. The apparently hard thing is to list them all, and to factorize non-prime numbers.
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I present to you: http://en.wikipedia.org/wiki/Generating_primes [wikipedia.org]
Twin Primes (Score:3, Funny)
Thanx xkcd! [xkcd.com]
Why do we believe the twin prime conjecture? (Score:2)
Do we have good reasons to think it's true? Or do we just see lots of twin primes and figure they never run out?
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The Question (Score:5, Informative)
Researchers hoping to get '2' as the answer
In case anyone's as confused as I was, I think I've finally figured out The Question, which is:
What is the smallest gap between consecutive primes which occurs infinitely many times?
Or something like that. Everyone thinks it's probably 2.
not sure... (Score:2)
Not doubting the guy's work, but I'm doubting the summary's "the gaps between consecutive numbers don't keep growing forever."
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It was already proved that there were an infinite number of primes.
Re:Open set it is! (Score:5, Informative)
Imagine that you did find all the prime numbers, every single one.
Then, take them, and multiply them all together.
Add 1.
You now have a number that is divisible by none of the primes, which therefore must be a prime number.
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You now have a number that is divisible by none of the primes, which therefore must be a prime number.
Or the number is divisible by a prime that wasn't in you initial set.
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Or the number is divisible by a prime that wasn't in you initial set.
GP has already used all the supposed finite number of prime numbers in constructing his contradictory bigger prime.
Re:Open set it is! (Score:5, Informative)
GP has already used all the supposed finite number of prime numbers in constructing his contradictory bigger prime.
The proof constructs a number that is not divisible by any of the prime numbers in the set of all prime numbers. Therefore it proofs there are an infinite number of prime numbers. The conclusion the constructed number must be prime is wrong.
Nyh
Re:Open set it is! (Score:5, Informative)
The GP's correction is right.
The GGP said that his number was prime. It might be, but it might not. But if it's composite then it cannot be divisible by any of the primes in his initial set so there must be a prime not in his set.
For example, if we assume 13 is the last prime then multiply them all together and add 1 we get 30031. But 30031 is not prime - it's divisible by 59 (which is a prime not in our set)
Tim.
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This whole proof is much easier if you use factorials as you can always prove there must be a prime bigger than N, as N! + 1 is not divisible by any number less than N. Which sort of gets around this weird way of attacking this problem y'all seem to be using which involves ephemeral 'set of known primes' which is weird in proofs.
Re:Open set it is! (Score:4, Insightful)
I don't see why it gets around this problem.
The equivalent claim would be that
N!+1 is prime.
The correct claim is that N!+1 is prime or is divisible by a prime larger than N
The faulty proofs are trying to construct a prime not in the set. The correct proofs are showing that a prime exists that is not in the set without making any claims about what that prime is other than it's bigger than N.
I'm pretty sure that it has been proved that there cannot be a constructive proof that there are an infinite number of primes - i.e. there is no way to construct a prime larger than N for arbitrary N.
Tim.
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I was just objecting to the use of set of all primes, as you say there is no easy way to construct (or test) primes, however by showing that there must be a prime greater than an arbitrary value you have demonstrated there are an infinite number of primes* without requiring that you know all the primes in the first place.
*(N!+1)! +1 ad infinitum.
The proof in the article is that there exists an infinite subset of the primes where members are separated from at least one other member by less than 70 million. W
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You don't have to know all the primes in the first place for the proof to work, you just have to postulate that such a finite set exists. (Which you then disprove by contradiction.)
Trivially constructive (Score:2)
N!+1 either is prime or has prime factors not in 1...N. Try factorizing the integers N+1 ... N!+1 in turn until you come to one that is prime.
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Why on earth would I do that?
PRIMES is in P. So if I want a prime bigger than N I'd start testing numbers of the form N+s for increasing s for primality until I found one. (Actually for very large values of N there are better candidates to test which have special case tests for primality that are particularly fast)
I wouldn't try to find a prime bigger than N by trying
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Oops, sorry, I misread your algorithm.
But it still doesn't help. You've assumed that it will terminate (i.e. there is a prime larger than N). So you cannot use it to prove that there is a prime larger than N.
Tim.
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I'm pointing out that the existing proof is constructive, not giving another proof.
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http://www.math.psu.edu/sellersj/courses/math035/fa11/handouts/07_infinitely_many_primes.pdf [psu.edu]
1) This proof is not a âoeconstructiveâ proof. We do not build an infinite list of primes in the process. This is a proof by contradiction.
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it provides the contradiction by constructing a list of numbers at least one of which must be a new prime. It's simple to test each of them. If you want to construct an infinite list of primes, just repeat indefinitely.
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Yes, Sorry. My comment wasn't very well written.
What I was trying to say is that given a purported complete set of primes, it's impossible to construct a prime not in the list other than by first assuming that there is a prime not in the list.
Any algorithm that tests N+1, N+2... will not terminate if N is the largest prime.
Tim.
Re: (Score:2, Insightful)
1) suppose you have a set of all primes, and the set is finite.
2) show that there's another prime not in your set - that contradicts (1).
3) therefore, there is no finite set that contains all primes.
All you've done is demonstrate one example of step 2. The original proof given by phantomfive gives a different example of a prime not in the set. Either works - the proof is valid.
Re:Open set it is! (Score:5, Insightful)
You've misunderstood the proof as a test to see whether a subset of primes up to prime n is complete. That's not the case. You start by taking the entire postulated finite set of primes.
The condradiction you receive - that it's possible to create a prime outside of the complete set of all primes - indicates that any finite set is incomplete. (Or alternatively that addition, multiplication, or sets work very differently than we assume, but let's stick to the form of mathematics the problem addresses.)
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It can't be composite if it's a product of all of the primes plus one. It could only be a composite if it was a product of a subset of the primes, plus one.
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It occurs to me that my comment below is rather demonstrating your point, actually. If it's a composite then we've successfully proven that the complete set of primes is an incomplete subset of the complete set of primes, which is another contradiction.
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You now have a number that is divisible by none of the primes, which therefore must be a prime number.
Or the number is divisible by a prime that wasn't in you initial set.
No, the assumption was that you have all prime numbers. You're not allowed to violate assumptions within a formal proof, you're only allowed to point out self-contradictions.
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No! Why is this causing so much confusion.
I claim that SEO (Some enormous number) is the largest prime.
You construct 2*3*5*7*11*...*SEO+1 and claim that it is a prime not in my list.
I run a quick probabilistic primality test and prove that your number is composite. (which it almost certainly is)
Conjecture: There are no numbers of the form 2*3*...*P_{n-1}*P_n + 1 that are prime for P_n greater than 11.
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The stated proof assumes the new number is prime. This is not a valid assumption.
Isn't it, at least under the stated conditions? The assumption that the new number is prime is made because it is not divisible by any of the supposedly complete set of primes used in its construction. I don't quite see why you need to add the "...or it is composite" to complete the reductio ad absurdum*.
*which is either what they told me proof by contradiction is called in Latin, or something Harry Potter was taught in Transfiguration.
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You now have a number that is divisible by none of the primes, which therefore must be a prime number.
Or the number is divisible by a prime that wasn't in you initial set.
The operation itself is guaranteed to give you a number that is coprime [wikipedia.org] with the initial set. However, if you were to believe that there were a finite set of prime numbers and were then to use that finite set as the input into the coprime generator, you'd get something that is coprime with "all" prime numbers, which would therefore consequently show that there must be at least one prime number that is not in that set, and establish the result as a candidate for the missing prime. (If you previously believed
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That clarification was important. GP said:
> You now have a number that is divisible by none of the primes, which therefore must be a prime number
This is incorrect. The number must have a prime factor not in the initial list, which is a different (and more general) statement than "it must be a prime number."
The existence of a prime factor not in the original chosen set is proof that the set was not, in fact, all the primes. Thus you've shown that the original premise leads to a contradition, so the origin
Re:Open set it is! (Score:4, Interesting)
Or more elegantly in haiku form:
Top prime's divisors'
product (plus one)'s factors are?
QED, bitches!
-- http://xkcd.com/622/ [xkcd.com]
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If you take all the primes from 2 to 23, and multiply them all then and one, you get 223092871 a non-prime, with 317 and 703763 [wolframalpha.com] as its prime factors.
I don't, however, see how it is obvious that multiplying all the primes in a list, then adding one, should mean that the result cannot be factorised by the original component primes.
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p is any of the primes, x is the result of multiplying all the other primes in the list.
If px+1 was divisible by p, then px+1 = py, for some whole number y
dividing by p, y = x + 1/p. This cannot be a whole number as p >= 2 and x is a whole number.
I'm sure there's a better proof, but that's just off the top of my head.
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If {2, 3, 5, 7, 11, 13, 17, 19, 23} are all the primes that exist (we know that they aren't, but just assume for the moment that they are), then the total 223092871 is indeed prime when compare
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The thread of comments attached to parent reminds me of folks on Yahoo Answers trying to apply order of operations to basic equations -- Don't look it up, humanity needs your hope.
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You should have explicated how the contradiction establishes that the original assumption must be wrong (i.e., the primes cannot be finitely listed). Not everyone understands proof by contradiction, so leaving it unstated was asking for trouble.
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No it wouldn't. 2 is a prime number. Any positive integer multiplied by 2 is an even number, thus the result of multiplying "all" primes is an even number. Any even number, plus 1, is not an event number. QED.
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Wrong. Multiplying all the known prime numbers will always give you an even number, so adding 1 to whatever result will give you an odd number.
Hint: 2 is a prime number. You figure out the rest.
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You should learn what multiply means before opening your mouth.
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The Greek knew in 300 BC there are an infinite number of prime numbers. The same proof also shows the gab arbitrary between two numbers can be arbitrary large (even larger as 70 million).
This proof shows there are an infinite number of primes that are 70 million or less from each other.
Nyh
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May. There is a trivial proof that there exist gaps larger than any given number ...
Pick any number n. Consider n! (that's "factorial", for the non-mathematicians). Now, n! - 1 might be prime (or not), but as n! is divisible by k for all k x and a prime q > p with q - p = 70 million, not that there will always be a prime within 70 million of x.
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I gather the comment system doesn't like all those symbols. It removed half of my reply. Let me try words ...
n! is divisible by k for all k less than or equal to n, so n! - k is divisible by k and (if k is not 1) is not prime. So n! - 1 to n! - (n + 1) are two numbers with a difference of n with no primes between them.
The result must show that for any x there are primes p and q with q > p > x and q - p less than 70 million, ...
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Nope, still with the missing stuff there it doesn't make sense :-)
But that might just be because I'm slightly blind when it comes to mathematical proofs.
(Kinda like Zaphods sunglasses, but build into the brain, blacking out when stuff gets complicated)
Re:Open set it is! (Score:5, Interesting)
From a purely mathematical point of view you are incorrect.
The proof isn't that there's less than 70million units between each prime (like there's a lot of primes with a gap of two units eg 29 and 31, 41 and 43 etc). the proof is that there's in infinite number of prime pairs with a maximum of 70 million units between them.
You can still find gaps significantly larger. Those gaps are present between numbers that are NOT prime pairs.
eg: 29 30 31 32 33 34 35 36 37 39 40 41 42 43 44
Here there is a prime pair with a 2 unit gap between them (41 and 43), however the number 37 has a larger gap on either side, because it is not a part of a "prime pair". In your thinking you are excluding the primes that are NOT paried, and the gaps between where one pair ends and another begins. Each of which, according to the proof still has the ability to exceed 70 million units.
Disclaimer: I did not fully read the proof posted in annals of mathematics, but I'm pretty certain that this is the gist of it
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Now add one to X. Now add one to X.
Oops. Just add one, not two ones. Two shalt thou not add.
Re:Open set it is! (Score:4, Insightful)
Your proof as written is wrong.
I claim there are a finite number of primes viz:
2 3 5 7 11 13.
You construct 2*3*5*7*11*13+1 = 30031 and claim that this is a new prime in my list.
I say - no it's not 30031 is composite. (59*509)
--
The correct proof is to say that X+1 is either prime or is divisible by a prime not in the list thus proving that the list is incomplete. If the list contains all the primes up to N then there must be a prime bigger than N.
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Actually, Euclid's proof [wikipedia.org] for the infinitude of primes says that the number itself is either a prime (which your example shows isn't always the case) or that the number can be factored by a prime not in the list provided (thus proving other primes exist). In your case both 59 and 509 are primes, showing the original list of primes was incomplete. Rinse and repeat.
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Oh god, I saw the "--" in your comment and assumed it was the signature... only then I realized it was the final section of your comment. Sorry about that.
Re:Open set it is! (Score:4, Informative)
Euclid's Theorem in actuality [clarku.edu] does refer to the case where X+1 is not prime. It's essential to the proof.
It goes something like this:
---------
Take a finite list of prime numbers, A, B, C etc. (The assumption that they are "all the primes" is unnecessary.)
Find the smallest common multiple of them, X.
Add 1 to that.
The new number, X+1, is either prime or composite.
If it's prime, then that's it. We've generated a new prime not on the list.
If it's composite, then it is divisible by some prime, G.
Could G be one the primes (A, B, C. etc.) already on the list?
But remember, X is divisible by A, B, C etc. So if G is one of those primes, then that means that both X and X+1 are divisible by prime number G, which is impossible.
Therefore G would have to be a new prime, not on the list.
Now we have a larger list, A, B, C, G, etc. and can repeat the process.
We can always generate a new prime not on the list, and therefore the list of primes is without bound.
---------
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That would make it 41?
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Re:'2' - wrong, its 42 (Score:5, Insightful)
Some AC felt the need to make a lame '42' reference. Then, against all odds, it somehow managed to get back around to being on topic when someone else gave it a -1, thus rendering it a nicely prime 41. Then you came along and decided to be an ass. Well done.
But wait! With 41 you don't just get an "on topic" prime number. You'll also find that 41 is actually a twin in the twin prime pair of (41, 43)! That's right, it is completely on topic... so.... nah nah nahnah nah.
Now, as far as I can tell I've managed to make two relevant posts on the topic out of a seemingly impossible "42 duh duh" comment. On the other hand, you've managed only to be an asshole and contribute nothing other than bad karma. As far as you comment about making more money goes, I'm confused, who knows, maybe I got whooshed or missed a meme or something. Or maybe I've just been trolled. But, maybe you'd make more if you weren't such an asshole and instead just let people have a good time without trying to piss on 'em. Especially when it doesn't even matter.
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You sound like the life of the party.
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He would be, but he doesn't get invited to those sorts of parties.
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Yeah, that movie right?
Saying 42 might have been funny if they were researching a number that had some abstract relationship to the meaning of life - but even then it would be predictable and overused.
But it's not funny just to answer 42 to any mathematics question. It's not funny at all.
42
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Exactly this.
Also, I was not in the mood to handle these parrotsheep who bleat out, "42! Get it?? Haha!" on cue, any time there is any remotely math-related discussion taking place.
Also, I see what you did there at the end, but I will not gratify it with a response.
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I'm not sure it does.
"there are infinitely many pairs of primes that are less than 70 million units apart"
It just means that the individual primes in the pairs must be 70 million units apart (from each other) or less. (and where the hell did "unit" come from? Do they mean integer?)
Not that single primes must be. Not that one pair from the next must be. You could have twin prime pairs at any interval so long as the other half of the pair is within 70 million integers.
Unless I'm reading it wrong. The th
Re:Primes closer together? (Score:5, Informative)
Not quite.
This means that for every prime p such that p+q (where q is less than 70 million) is also prime, there exists another prime r bigger than p such that r+s (where s is also less than 70 million) is also prime. Note that there is no limit to the distance between p and r.
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Yes, it's more like this: Imagine if you took a sack of marbles and spread them infinitely thin, you'd expect that the distance between any two marbles to also grow to infinity. This is proof that primes are not like this, no matter how thin they're spread they'll cluster in pairs less than 70 million apart. The conjencture is that you'll always find another pair 2 units apart (like 5 and 7, 11 and 13 etc.) no matter how big the numbers get.
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Yes, it's more like this: Imagine if you took a sack of marbles and spread them infinitely thin, you'd expect that the distance between any two marbles to also grow to infinity. This is proof that primes are not like this, no matter how thin they're spread they'll cluster in pairs less than 70 million apart. The conjencture is that you'll always find another pair 2 units apart (like 5 and 7, 11 and 13 etc.) no matter how big the numbers get.
It would of course depend on _how many_ marbles there are and _how thin_ they are spread. In the case of prime numbers, there are still so many of them that we _expect_ two that are close together from time to time.
The number of primes If you pick a random integer around N, the chance that it is a prime number is about 1 / ln (N). If you pick an odd number, the chance is about 2 / ln (N). Now if you pick an odd number x, then the chance that x is prime is about 2 / ln (N), the chance that x + 2 is prime
Re: (Score:2)
The number of primes less than N is about N / ln (N).
If you pick a random integer around N, the chance that it is a prime number is about 1 / ln (N). If you pick an odd number, the chance is about 2 / ln (N).
Now if you pick an odd number x, then the chance that x is prime is about 2 / ln (x), the chance that x + 2 is prime is also about 2 / ln (x), both are not quite independent (if x is not divisible by 3, then x + 2 is more likely divisible by 3, same for 5, 7 etc. ), but the chance
Re:TFS (Score:5, Funny)
This is probably the worst written summary that I have ever read on Slashdot.
You must be new here.
Re: (Score:3)
Not only can there be increasingly large gaps but there are increasingly large gaps.
(N+1)!+2 to (N+1)!+N+1 are N consecutive composite numbers - divisible by 2..N+1 respectively.
Therefore there are arbitrarily long sequences of composite numbers.
Tim.
Re: (Score:2)
Have you considered the possibility that I might be replying to a particular comment in the parent my post was attached to?
The comment in particular that I was replying to had:
"but it's not true for every prime number, so there can still be increasingly large gaps."
To which my reply said:
"Not only can there be increasingly large gaps but there are increasingly large gaps."
Tim.
Re: (Score:2)
means that that the gaps between consecutive numbers don't keep growing forever.
What I see now is that it really means that while gaps can (and presumably do) keep to a general trend of growth, there will always be a gap of less than 70 million somewhere up ahead.
Re: (Score:2)
Nope, it is true. Consider only the prime pairs. There are, we now know, no gaps larger than 70 million. Add the rest of the prime numbers and there can still be no gaps larger than 70 million.
Blatant nonsense. Not only has someone already posted how to find 70 million consecutive non-primes, but it has been long proven that the number of primes less than N is about N / ln (N) (the ratio of number of primes = exp (70,000,000) the _average_ gap between primes is more than 70,000,000.
Re: (Score:2)
finds that there are infinitely many pairs of primes that are less than 70 million units apart.
Doh. I read "pairs of primes" as "twin primes" - as in pairs of twin primes less than 70 million apart.
Re: (Score:2, Offtopic)
good to know that while the country is almost 17 trillion in debt that such important endeavors take priority
What do you mean - they can now say "17 trillion dollars in debt is close to two dollars in debt when you consider infinity, the mathematicians say so!"
Re: (Score:2)