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Major Advance Towards a Proof of the Twin Prime Conjecture
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."
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.
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.
Stories like this only remind me of how ignorant I still am and how I've wasted my life.
To be perfectly honest the proof that the gap between consecutive integers doesn't grow forever is pretty simple. It stays 1.
It was already proved that there were an infinite number of primes.
There is a simple, ancient, proof that there are infinite prime numbers.
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.
"First they came for the slanderers and i said nothing."
That would make it 41?
That there are infinitely many pairs of prime number which are at most 70 million numbers apart does not guarantee that you'll find another prime number within 70 million from a given prime number. There are (infinitely many) prime numbers for which you'll find another one within 70 million, but it's not true for every prime number, so there can still be increasingly large gaps.
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
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.
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 thing about maths is, you have to be VERY precise and not leap into assumptions without testing them very, very, very thoroughly first. 99.99999% certain isn't good enough for a mathematician.
That was the point, yes. Are you unclear on the nature of the proof?
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.
systemd is Roko's Basilisk.
http://primes.utm.edu/notes/proofs/infinite/euclids.html
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.
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.
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
Now add one to X. Now add one to X.
Oops. Just add one, not two ones. Two shalt thou not add.
systemd is Roko's Basilisk.
This is probably the worst written summary that I have ever read on Slashdot.
You must be new here.
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.
To be pedantic, we still cant conclude the product + 1 is prime, only that it is a contradiction that it is divisible by no prime (which is all we need anyway). The GP is correct in that a proof more similar to Euclid's original is given by considering an arbitrary finite set P of primes, letting N be the product of the P plus 1, and then concluding that a prime divisor of N must not be in P.
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.
God said, "div D = rho, div B = 0, curl E = -@B/@t, curl H = J + @D/@t," and there was light.
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.
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, ...
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.
God said, "div D = rho, div B = 0, curl E = -@B/@t, curl H = J + @D/@t," and there was light.
This isn't pedantic at all. The constructed number does not have to be a prime (as shown below in locofungus' example) and thus the additional step (there must be another prime, contradicting the assumption) is needed.
There is an easy proof that there can be arbitrarily large gaps between consecutive primes:
For example, suppose you want to find a prime gap of size at least 1000000. Let N = 1000000!, so, ..., N + 157 isn't prime because it's divisible by 157, ..., N + 1000000 is divisible by
in particular, N is divisible by every integer from 2 to 1000000. Then N + 2 isn't prime because it's
divisible by 2,
1000000. So there are 999999 consecutive composite numbers from N + 2 to N + 1000000. The
gap between the largest prime before them and the smallest prime after them is at least 1000000.
What the new proof shows is that as we go through the primes, the prime gaps aren't eventually
all huge: there will always be some further prime gaps that are smaller than 70000000.
This is perhaps more impressive when you realize that the Prime Number Theorem says that
the average size of the gaps does get as big as you like: for billion-digit numbers, the
average size of the gap between primes is about 2.3 billion. Nevertheless, close pairs of primes
will still occasionally occur.
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.
"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).
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 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 that there are no prime numbers, the product-over-set-and-add-one operation produces 2.) The assumption that must have been false, given that everything else is a basic mathematical or logical operation, is that there is a finite set of prime numbers; there must be an infinite number of primes.
But TFA wasn't talking about this. It was talking about the number of pairs of primes where the difference between the pair is 2, and that's a very non-trivial property.
"Little does he know, but there is no 'I' in 'Idiot'!"
Not quite.
What he proved is that given any number n, there exist prime numbers p and q that are (1) greater than n and (2) within 70 million of each other. For example, for a particular n, perhaps p = n + 1,000,000,000 and q = n + 1,069,000,000.
This is not quite the same as saying there's necessarily a prime between n and n + 70,000,000. In fact, since the average density of primes decreases as 1 / log(n), I'm pretty sure that statement is known to be false.
His proof is fine, you're just not familiar enough with how proofs by contradiction work. It's true that 30031 is composite. Within the formal context of his proof by contradiction, it is simultaneously true that 30031 is prime. It is prime because it is not divisible by any other prime. It is composite because you found a non-trivial factorization. Both statements are true. That's why it's a proof by contradiction - there is a contradiction. Your wording is correct too, but it is not necessary to add the case that you did where X+1 is not prime, though that extra bit may be helpful for less mathematically experienced readers.
Can people stop trying to point out the flaw in this comment please? The poster is correct and if you look up the proof you will find this rider.
You, sir, are not a mathematician. The person you are replying to is because he gets why the stated proof was incomplete.
If the new number is not divisible by any of the primes in the initial set, then either it is a prime (contradiction - the initial set is not complete) or it is composite (contradiction - another prime exists outside the initial set).
The stated proof assumes the new number is prime. This is not a valid assumption.
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!"
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.
God said, "div D = rho, div B = 0, curl E = -@B/@t, curl H = J + @D/@t," and there was light.
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.
So you don't have the set of all primes after all... that's the point. The proof goes like this:
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.
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)
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.
God said, "div D = rho, div B = 0, curl E = -@B/@t, curl H = J + @D/@t," and there was light.
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.
Live today, because you never know what tomorrow brings
Or more elegantly in haiku form:
Top prime's divisors'
product (plus one)'s factors are?
QED, bitches!
-- http://xkcd.com/622/
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. Which is a pretty hard thing to even get close to proving.
Actually, Euclid's proof 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.
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.
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
It's as much my fault. I'll try not to do that again!
God said, "div D = rho, div B = 0, curl E = -@B/@t, curl H = J + @D/@t," and there was light.
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.)
No kidding!!! What do you say at this point?
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 is about 2 / ln (N), 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 that (x, x+2) is a twin prime pair is roughly 1 / (ln (x))^2.
The sum of (1 / ln (x))^2 over all even integers x diverges; if you sum it for all x The same is true for every pattern like (x, x+2, x+6), or whatever pattern you choose, except for those patterns where it is obvious that these primes can't exist, like (x, x+2, x+10), where one of the three numbers must be divisible by 3.
Proving it is of course an entirely different matter. However, if there are infinitely many pairs of primes with a gap
So if you take the (x, x+2) conjecture aka twin prime conjecture, the (x, x+4) conjecture, the (x, x+6) conjecture and so on, which are _all_ assumed to be true, then we now know that at least one of them _is_ indeed true.
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 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.
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.)
No kidding!!! What do you say at this point?
Bloody html.
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 that (x, x+2) is a twin prime pair is roughly 1 / (ln (x))^2.
The sum of (1 / ln (x))^2 over all even integers x diverges; if you sum it for all x less than N then the sum is more than N / (ln (N)^2, which diverges.
Something similar is true for every pattern like (x, x+2, x+6), or whatever pattern you choose, except for those patterns where it is obvious that these primes can't exist, like (x, x+2, x+10), where one of the three numbers must be divisible by 3.
Proving it is of course an entirely different matter.
However, if there are infinitely many pairs of primes with a gap less than 70,000,000 then at least one of the possible gaps must come up an infinite amount of times (because 70 million times a finite number is still finite).
So if you take the (x, x+2) conjecture aka twin prime conjecture, the (x, x+4) conjecture, the (x, x+6) conjecture and so on, which are _all_ assumed to be true, then we now know that at least one of them _is_ indeed true. There is a K 70,000,000 such that there are infinitely many pairs of primes (x, x + K).
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.
No kidding!!! What do you say at this point?
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
--- To err is human... Am I more human than most ?
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.
No kidding!!! What do you say at this point?
You really don't get it, do you? And you tell other people that they're not familiar enough with how proofs by contradiction work...
Thanx xkcd!
On the one hand you take life too seriously, and on the other, you do not take playful existence seriously enough. Seth
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.
Get free bitcoins: http://freebitco.in
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?
Democracy Now! - your daily, uncensored, corporate-free
You sound like the life of the party.
I am very small, utmostly microscopic.
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.
Depends on what you mean by "construct". You can just start at N+1 and test everything for primality. We know now that primality testing is in P, and the prime number theorem tells us that the distribution of primes is dense, so this is even efficient.
You mean multiply? If you add then you get ambiguous cases, i.e 2+3=5 where you don't know if it was just five or if it was two and three. If you want to add you would have to do powers of two. Pretty cool, but it has nothing to do with the article :D
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.
God said, "div D = rho, div B = 0, curl E = -@B/@t, curl H = J + @D/@t," and there was light.
He would be, but he doesn't get invited to those sorts of parties.
Out of modpoints but really liked a post? 1BDkF6TtmmeZ3yqXbz9yhdYVqRYnwFoXDj
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.
systemd is Roko's Basilisk.
Ok, this isn't rigorous at all (obviously), but it seems to me that if the size of the gap continuously grew, but fluctuated randomly, you would still have an infinite number of primes close together even though the average distance between them never stopped increasing. They would become fewer and fewer, but never stop, and hence would be infinite.
Not doubting the guy's work, but I'm doubting the summary's "the gaps between consecutive numbers don't keep growing forever."
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.
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 formula_3, towards infinity ;
and plot two fomulas that offset formula_3 by formula_4 in both 'directions' ;
then Doesn't it make a lot of sense?
I'm terribly sorry for my bad mathmetics and would welcome a rant in which someone would explain why and where I'm making a mistake.
Here be signatures
Your math is correct. What I believe is being discussed is that while multiplying the primes from 2 to 23, you're making the (admittedly invalid) assumption that the primes from 2 to 23 are all that exist. We know that there are more primes than that, we're intentionally making that assumption to prove a point...
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 compared to all of the primes that "exist", thus invalidating the original assumption and proving it false because now we have another known prime. If you count the fact that 317 and 703763 are indeed prime factors, this also invalidates the original assumption in that there are primes in addition to all that "exist", also proving the assumption false.
Given that numbers are infinite, there would be no point in which you could truthfully say "here is the set of all existing prime numbers", thus proving that there are an infinite number of them. That's all that's being said by the proof posted so far above: There are an infinite number of prime numbers, and here is my proof as to why that is so.
Euclid's Theorem in actuality 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.
---------
There are two kinds of people: 1) those who start arrays with one and 1) those who start them with zero.
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.
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.
We know where leadership by an anti-intellectual "strongman" who scapegoats minorities and likes boisterous rallies goes
Which is... back on topic... prime!
My present is the activity I am currently engaged in with the purpose of turning the future into a better past.
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.
systemd is Roko's Basilisk.
Thanks for the tip.
"First they came for the slanderers and i said nothing."
You should learn what multiply means before opening your mouth.
I will take your advice, but it is interesting because just last night I was reading Dijkstra who said that in the Netherlands, order of operations puts multiplication before division.
"First they came for the slanderers and i said nothing."
No!
The original proof is flat out wrong.
It claimed that the product of all the primes up to N and then adding 1 is prime.
The product of all the primes up to 13 is 30031.
2^30030 mod 30031 is 21335
Therefore 30031 is not prime. (Fermat's little theorem)
The proof as stated is insufficient. I have proved that 30031 is not prime but I have not found any prime factors of 30031. Therefore, to complete the proof you need to include the case that 30031 contains a prime factor not in my original list because I have proved that 30031 does not belong in my list.
Tim.
God said, "div D = rho, div B = 0, curl E = -@B/@t, curl H = J + @D/@t," and there was light.
If there is a 70m gap between prime pairs, then there is a 70m gap between primes. Prime pairs are made up of primes. So it effectively proves that every series of 70m numbers has at least 2 primes and 1 prime pair. Is there a proof that narrows basic primes to a smaller maximum?
I only use irrational numbers as there are fewer rational numbers.
Well, yes, that's exactly what this is. Which is what the summary says.
systemd is Roko's Basilisk.
I didn't think that "there's an infinite number of twin primes" was in question. Like so: Let S(k) be the set of all prime numbers less than or equal to k, where k is at least 3 [S(3) would be {2, 3}]. Let P be the product of them all. Then P+1 and P-1 are prime and not in S(k). Therefore S(P+1) is at least two larger than S(k) (and probably much more than 2; finding those others, that's the tricky part). Rinse, repeat. What part of this is difficult?
Yes, but if you read the article, or hell, even the summary, then you'd know it was about primes.
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.
My present is the activity I am currently engaged in with the purpose of turning the future into a better past.
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 original premise is impossible.
Hmmm, I guess you must mean "for n greater than 11" (not "P_n greater than 11").
We know where leadership by an anti-intellectual "strongman" who scapegoats minorities and likes boisterous rallies goes
If the set of primes is finite, form the product of all primes and add 1, creating a number not divisible by any prime (making the number formed prime by definition) yet not included in the set of all primes by construction. At this point you can smoke some weed or you can begin to suspect that the set of primes is not finite.
Let p = 10^-googolplex.
With enough patience, you can win this lottery 100 times in a row, and you can do that as many times as you like.
All this new result gives us is further evidence that in the unextinguished coincidence of short spacings, the distribution of primes resembles a random process. There's a structural reason why both N and N+1 are never prime at the same time. It appears, however, to be rather difficult to identify any other structure of the distribution of primes taking the form of permanently extinguished gap distances.
Our list of viable gaps grows thin. (I did wish momentarily to mark that up as <e>thin</e> for Elvish italic.)
No, you are not getting it. Suppose that P[i] denotes the i-th prime number. The proposed theorem does not say that P[i+1]-P[i] <= 70 million for all values of i, as you believe.
What it does say is that for every index k for which P[k+1]-P[k] > 70 you can find a number m greater than k such that P[m+1]-P[m] <= 70 million.
Man, it kills me when someone quotes a great mathematician/great proof and the neckbeards come out to tell you that you are wrong.
I made the mistake of describing cardinality to a coworker and had him unwittingly suggesting that Cantor is wrong, he thought he was arguing with me.
Its all a big bologna measuring contest.
This sig is not paradoxical or ironic.
I did. I've also now tested it more fully and I think it's wrong.
The product of the first 75 primes (up to 379) is X=1719 620105 458406 433483 340568 317543 019584 575635 895742 560438 771105 058321 655238 562613 083979 651479 555788 009994 557822 024565 226932 906295 208262 756822 275663 694110
2^X mod X+1 is 1 so X+1 is probably not prime.
Tim.
God said, "div D = rho, div B = 0, curl E = -@B/@t, curl H = J + @D/@t," and there was light.
Gah. ... so X+1 is probably prime.
God said, "div D = rho, div B = 0, curl E = -@B/@t, curl H = J + @D/@t," and there was light.
Sometimes I get too far out with my humor, I apologise for making you feel that way.
is.41.aprimenumber.com has something stupid like that about a lot of numbers.
Is 1563649 a prime number?
Wolfram Alpha agrees: says it's a prime number.
We know where leadership by an anti-intellectual "strongman" who scapegoats minorities and likes boisterous rallies goes
Hm. I am not a math major nor a history major. How does multiplying prime numbers and adding 1 create a guaranteed prime number? I can think of smaller examples of multiplying primes and adding 1 that does not equate to a prime: (3*3)+1=10.
"Someone needs to talk to the tree of liberty about its ghoulish drinking problem." by ohnocitizen
It only guarantees that you will have a number that is not divisible by any of the numbers you multiplied together. Thus if you multiply every prime number in existence together, and add 1, you will end up with a number indivisible by every prime number in existence.
"First they came for the slanderers and i said nothing."