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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."
70 millionth post!
captcha: lessens
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.
They should not be overused because they are a limited mathematical resource, like badgers or squid. I take care to use mostly non-prime numbers, especially even numbers because they are renewable. It is a little thing we can all do to protect our Nation's planet and its numbers for future generations of Americans and maybe for Filipinos too.
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everyone knows the answer is 42
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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.
That means that prime numbers are an open set, since an infinite number of prime pairs with a finite difference means an infinite number of prime numbers.
In other words: If you wanted to be recognized for finding the highest prime number, you can stop your computer now. There is no highest prime.
But if you only care for a temporary entry, go ahead; you may well find one in a search of only 70 million numbers!
This is probably the worst written summary that I have ever read on Slashdot.
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.
Also means that there must be at least one prime in every sequence of 70 million integers.
Means I can put an upper bound on my prime search script....If it searches 70,000,001 consecutive integers and claims to have found no primes, I'll know the bugged little script is lying.
That's a helpful debugging heuristic. Thank you, Pure Math.
I've been using a similar practice for a few years when implementing the API database for TT Livescan.
http://tot-ltd.org/API/
http://tot-ltd.org/API/main.db
Use prime numbers 2 or higher to map API calls to a "number" specific family (add the collective values of the API calls from main.db, then convert the value to hexadecimal), based on the API functions (Windows 3.11 to Windows 7). The rate at which it can catch malware based on API calls alone is grotesquely efficient.
"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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Thanx xkcd!
On the one hand you take life too seriously, and on the other, you do not take playful existence seriously enough. Seth
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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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.
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."
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
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.
I did not read the paper, but the statement above doesn't mean that gaps between consecutive groupings of prime numbers won't keep growing forever.
It's basically like saying there are an infinite number of "star pairs" in the universe that are less than 70m light-years from each other. That doesn't mean that the spacings between these star-groups (let's call them "galaxies") isn't getting larger over some way of measuring (let's call that way "age of the universe").
Perhaps the paper does make some guarantees in this area that aren't made in the simplified /. summary.
Knowledge is how to play a game, intelligence is how to win, wisdom is knowing what game to play.
We're half-way there!
Rather proving the twin prime conjecture (which is only for the case concerning prime numbers with a gap of n=2), is it more likely it is the first proof for any case of Polignac's conjecture (n = 70 million)? A large step though to proving the special case of twin primes.
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?
You are only exchanging one problem for another.
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.)
You are not smart.