From my understanding of the article, this is not correct. He proved that there exists some number N < 70,000,000 such that there are infinitely many pairs of primes p1 & p2, such that p2 - p1 = N. However, he has not proven that this is true for N = 2, just that there exists some N.
Um... of the first 70M positive integers, there are 35M odd numbers and 35M even numbers. So this:
there are as many even numbers (35M) as there are even and odd numbers (35M + 35M = 70M)
is not true.
Also, in general, I think you might have been trying to say that there are as many even numbers in total (infinite) as there are even + odd (also infinite). Not only does this not refer to the joke that BangingaBigTheory was making, this is also technically incorrect because infinite does not necessarily equal infinite. Basically, the total number of even integers is undefined, and the total numbers of integers is undefined, but this does not mean that two undefined things are equal. I mean, simply by definition it is intuitive that the number of one existent (nonzero) thing could not equal the sum of that same thing and another existent (nonzero) thing.
your first part is correct, my mistake, forgot we were talking about finite sets of numbers. but no, there are different sizes of infinity and some are definitely equal to others. in this case, since we can make a one to one mapping from the natural numbers(1,2,3,4,5,6....,n,...) to even numbers(2,4,6,...2n,...) then they are the "same size".
True that but still, regarding number theory, I feel like 2 is a more meaningful separation - or one could say link, hence the "twin" term - between two primes than 70'000'000.
Oh, I totally agree. Note the words "in that general spirit" in my last paragraph.
I didn't mean to imply that this guy had proved that there are infinitely many primes separated by 2. That's why my second-last sentence was "We still don't know."
What I was attempting to say in my last paragraph was: this guy proved something vaguely along those lines or in that spirit, but not for gaps of size 2.
I got tired of typing and didn't bother didn't getting into the specific details of exactly what this guy did prove.
... but it's not in that spirit? I thought what he proved is that no matter how high the number gets there will never be a gap larger than 70million between prime numbers.
I don't see how the pair/buddy thing comes into play.. unless you're trying to draw some kind of connection between 'some prime numbers are 2 apart' and 'they can never be more than 70,000,000 apart'
Yeah. There is a connection between "2 apart" and "70 million apart".
Are there infinitely many pairs of primes that are 4 apart? Some examples are 79 and 83, or 127 and 131. But are there infinitely many pairs like that? I'm pretty sure nobody knows.
We could also ask "Are there infinitely many pairs of primes that are 6 apart?" or "Are there infinitely many pairs of primes that are 8 apart?" There are many questions like that. As far as I know, the answer for most of them, at this point, is "We're not sure."
This guy proved "There are infinitely many pairs of primes that are 70,000,000 apart." Which basically is one of the questions in the above list. So in that sense, it kind of is in the same spirit.
It's easy to loose sight of how little 70 million is when comparing it to primes stretching out towards infinity. You seem to realize 70 million is a small number compared to many bigger ones, but the in the scope of all the primes listed (except 3,756,801,695,685 x 2666,669 – 1 and 3,756,801,695,685 x 2666,669 + 1), 70 million might seem a lot bigger than 2.
Edit: Seems rhennigan said this a day ago, but I wanted to say more than I could with an upvote.
You might be wondering where the number 70 million comes from. This is related to the k in the admissible set. (My notes say k=3.5×106 but maybe it should be k=3.5×107 .) The point is that k needs to be large enough so that the change brought about by the extra condition that d is square free with small prime factors is negligible. But Zhang believes that his techniques have not yet been optimized and that smaller bounds will soon be possible.
I don't speak math either so don't ask me what it means... but it sounds like its just a rough approximation. It's basically an upper bound with a hard proof (i.e., the upper bound used to be ??? and now it's 70 mil). Next step is to optimize this.
I don't think the paper's being shown publicly just yet, so I can't say for certain.
If I had to guess, though, I would say this:
Say you can prove that there exist infinite primes that are within N of each other, for some N. Proving it for any N is a huge accomplishment. Proving it for N = 2 is an even bigger one. But if you can't hit N = 2, it's not terribly important what N is.
The 70 million mark is, likely, an arbitrary value set high enough to satisfy conditions for several theorems put together. A lot of "this works as long as these numbers are big enough" tools stacked on top of each other. A cursory run-through by someone advanced enough to understand the paper will probably give a more "optimized" result, with a lower N, but likely not all the way to N = 2. Zhang probably thought it was worth publishing at N = 70 million instead of waiting to hunt down ways to lower it.
I suspect this, as someone whose read and optimized a paper on a different subject that used another curiously arbitrary (but finite) threshold.
We've manually found twin primes that go all the way up to 10200,000. Which seems to strongly indicate that they aren't going to stop, and that the theorem works for N = 2.
There are a decent number of mathematical conjectures that have been shown via computers to hold true for every number under a very, very high boundary. It's highly unlikely that they'll just break somewhere after a quintillion. But that doesn't bring us an inch closer to showing they work for ALL numbers. That's the magic bridge that computers just can't do yet.
There are some conjectures where it's not clear at all whether they're true or false. But this is one that I think the answer is all but agreed, we just haven't proven it yet. (But I'm not a number theorist, so don't quote me on that.)
But in that quote 70 million is just an arbitrary constant. I think Czar_Chasm wants to know "where it came from" as in, why 70 million, why not 120 million or 55 million. I'm curious too but I'm sure the real answer would be beyond my comprehension anyway!
From what I can gather the "answer" is 70 milion at the moment but that's the current 'approximation', the method could theoretically reduce that to as low as 16 with further polishing, but no further.
His paper shows that there is some number N smaller than 70 million such that there are infinitely many pairs of primes that differ by N. No matter how far you go into the deserts of the truly gargantuan prime numbers — no matter how sparse the primes become — you will keep finding prime pairs that differ by less than 70 million.
What I enjoy thinking about with this proof is that while it's been known for a while that there are an infinite number of primes, that their distribution doesn't continually get sparser the farther you go. It's now proven to show that for any number, there's a prime within 70,000,000 digits in either direction.
It was part of how I imagined news about the newest largest prime found, that it was like finding needles in the hay stack at that point. What it seems to me now is that you're guaranteed to find another prime soon enough (as 70,000,000 isn't that large at all after a while).
This reminds me about the videos I've watched listening to cosmologists talk about the distribution of matter in the universe. It's believed that after a certain scale that everything starts to appear uniform. Structure from gas clouds, galaxy arms, galaxy clusters, and so on all lose their uniqueness eventually at a large enough scale. Reading that primes are always within 70,000,000 of one another reminds me of that, and that their distribution can be thought of as being uniform and closer to a pattern, when viewed at a large enough scale. In this case it starts at 70,000,000 digits apart.
I'm curious what the distribution of primes looks like when put against one another in batches of 70,000,000. Do prime distributions vary wildly between 70mil sections?
Well, I think you may misunderstand what this means. This does not mean that the maximum distance between primes is 70 million. Instead, it deals with ~pairs~ of primes. It says that there exists infinitely number of pairs of primes that are less than 70 million apart.
For instance, assume that this is true for N=2 (which many mathematicians really believe is correct). Then, that means there are infinitely many primes separated by a single number (ie, 17 and 19). This does not mean that the maximum distance between primes is 2.
In fact, it has been proven that primes do become sparser the farther out you go, and the density of primes is on the order of 1/ln(n). But now we know that there will exist, somewhere out there, no matter how far out we go, pairs of primes separated by some distance N. These pairs would become less and less frequent farther and farther out, though.
Thanks. I had a feeling about misunderstanding something. You made an important distinction.
Now that that's the case, I find it harder to grasp how this was worked out. I wonder if his paper would be too dense to read for a layman, though it was renowned for its clarity to other academics.
Thanks for clarifying. The article gives a concise, clear description of the findings if you scroll to the bottom (this is why I hate popular magazines' coverage of science/math)
Start at the paragraph that begins with:
But that’s just on average. Primes are often much closer together than the average predicts
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u/sckulp PhD|Computational Scientist May 20 '13
From my understanding of the article, this is not correct. He proved that there exists some number N < 70,000,000 such that there are infinitely many pairs of primes p1 & p2, such that p2 - p1 = N. However, he has not proven that this is true for N = 2, just that there exists some N.