# "Where is the door?"January 27, 2015 8:29 AM   Subscribe

Profile: Breaking down the problem of bound gaps [New Yorker]: After graduating with a Ph.D. in algebraic geometry from Purdue in 1991, Yiting Zhang kept the books for a friend's Subway franchise and found other odd jobs before taking up a part-time calculus teaching position at the University of New Hampshire in 1999.
“For years, I didn’t really keep up my dream in mathematics,” he said.

“You must have been unhappy.”

He shrugged. “My life is not always easy,” he said.
He published one paper in 2001. Then, in 2013, he submitted "Bounded Gaps Between Primes" to Annals of Mathematics, one of the most prestigious journals in the field, which contained a proof for a finite bound within which there exist an infinite number of pairs of primes. It was a stunning mathematical breakthrough.

An excerpt on the many interesting types of primes:
Prime numbers have so many novel qualities, and are so enigmatic, that mathematicians have grown fetishistic about them. Twin primes are two apart. Cousin primes are four apart, sexy primes are six apart, and neighbor primes are adjacent at some greater remove. From “Prime Curios!,” by Chris Caldwell and G. L. Honaker, Jr., I know that an absolute prime is prime regardless of how its digits are arranged: 199; 919; 991. A beastly prime has 666 in the center. The number 700666007 is a beastly palindromic prime, since it reads the same forward and backward. A circular prime is prime through all its cycles or formulations: 1193, 1931, 9311, 3119. There are Cuban primes, Cullen primes, and curved-digit primes, which have only curved numerals—0, 6, 8, and 9. A prime from which you can remove numbers and still have a prime is a deletable prime, such as 1987. An emirp is prime even when you reverse it: 389, 983. Gigantic primes have more than ten thousand digits, and holey primes have only digits with holes (0, 4, 6, 8, and 9). There are Mersenne primes; minimal primes; naughty primes, which are made mostly from zeros (naughts); ordinary primes; Pierpont primes; plateau primes, which have the same interior numbers and smaller numbers on the ends, such as 1777771; snowball primes, which are prime even if you haven’t finished writing all the digits, like 73939133; Titanic primes; Wagstaff primes; Wall-Sun-Sun primes; Wolstenholme primes; Woodall primes; and Yarborough primes, which have neither a 0 nor a 1.
Previously on Metafilter.
posted by ilicet (67 comments total) 43 users marked this as a favorite

a finite bound within which there exist an infinite number of pairs of primes

O.K., that sounds like a flat contradiction. I assume what it means is that there is some (indeterminate) range which no matter where you place it on the infinite number line will contain at least one twin prime pair? In other words, that no matter what integer you start at, there is some maximum finite number of subsequent integers you would have to count through before hitting upon a twin prime pair? Is that right?
posted by yoink at 8:51 AM on January 27, 2015 [2 favorites]

Yeah, if you hit the previously the result is defined much more clearly there.
posted by kmz at 8:54 AM on January 27, 2015 [1 favorite]

Cool, thanks for posting!
posted by internet fraud detective squad, station number 9 at 8:55 AM on January 27, 2015

Ah, sorry I worded that badly--but yeah, as I understand it Zhang proved that a finite bound of 70 million, placed at any part of the infinite number line, will contain a pair of primes. Other mathematicians have since then narrowed down the size of the finite bound. (Not necessarily twin primes two apart, just that there are two primes.)
posted by ilicet at 9:05 AM on January 27, 2015 [1 favorite]

The gap is down from 70 millions to 246

...Tao had the idea for a coöperative project in which mathematicians would work to lower the number rather than “fighting to snatch the lead,” he told me.
The project, called Polymath8, started in March of 2013 and continued for about a year. Incrementally, relying also on work by a young British mathematician named James Maynard, the participants reduced the bound to two hundred and forty-six. Having discovered that there is a gap, Zhang wasn’t interested in finding the smallest number defining the gap. This was work that he regarded as a mere technical problem, a type of manual labor—“ambulance chasing” is what a prominent mathematician called it. Nevertheless, within a week of Zhang’s announcement mathematicians around the world began competing to find the lowest number. One of the observers of their activity was Terence Tao, a professor at U.C.L.A. Tao had the idea for a coöperative project in which mathematicians would work to lower the number rather than “fighting to snatch the lead,” he told me.
The project, called Polymath8, started in March of 2013 and continued for about a year. Incrementally, relying also on work by a young British mathematician named James Maynard, the participants reduced the bound to two hundred and forty-six.

Excellent article.
posted by francesca too at 9:08 AM on January 27, 2015 [5 favorites]

I read an interview once with a famous poet (wish I could remember who) who said he loved reading about horse racing because the terminology and names were so musical. I feel the same way about mathematics. "Sexy primes," perfect.
posted by sallybrown at 9:10 AM on January 27, 2015 [1 favorite]

"Sexy primes," perfect.

Perfect name for a Derby winner, you mean?
posted by deludingmyself at 9:12 AM on January 27, 2015 [6 favorites]

In the list above, I'm having difficulty wrapping my head around curved-digit primes (0, 6, 8, and 9) and holey primes (0, 4, 6, 8, and 9). Aren't these primes simply by-products of the random lines used in the writing system that transcribes them--in this case, the Arabic numeral system--rather than numbers with legitimate properties that might be further investigated? Transcribed into Japanese (curved primes: 零・六・八・九）or, for that matter, into any non-base-ten numbering system, curved primes and holey primes become run-of-the-mill, humdrum, nothing-more-than-meh primes.

Are there any characteristics of curved-digit or holey primes that might be described as anything other than an absurdist joke?
posted by Gordion Knott at 9:13 AM on January 27, 2015 [1 favorite]

Wait, so Terence Tao et al showed there must be at least two primes in any set of 246 consecutive positive integers? (I may have an off by one error, sorry, it's early).

That's astonishing.
posted by leahwrenn at 9:20 AM on January 27, 2015 [1 favorite]

Not quite. There are an infinite number of pairs of primes no more than 246 apart from each other. That doesn't mean there's always a pair of primes in any 246 number interval.
posted by kmz at 9:28 AM on January 27, 2015 [15 favorites]

Remember that this is a weaker form of the Twin Primes conjecture, which says that there's an infinite number of primes two away from each other. Obviously there's not always two primes in any set of 3 consecutive integers.
posted by kmz at 9:31 AM on January 27, 2015 [2 favorites]

That doesn't mean there's always a pair of primes in any 246 number interval.

And indeed the interval 247!+2, 247!+3, ..., 247!+247 contains no primes.
posted by gleuschk at 9:35 AM on January 27, 2015 [18 favorites]

And yeah, you can find a gap of any length between primes, as neatly shown above.
posted by kmz at 9:37 AM on January 27, 2015

you can find a gap of any length between primes

Doesn't the FPP directly contradict that statement?
posted by yoink at 9:46 AM on January 27, 2015

OK help me out here....247! is obviously not prime......how do we know that 247! + 2 is not prime?

Is it simply that 247! + 2 can be factored into 2*(247!/2 + 1)? that seems legit......
posted by thelonius at 9:47 AM on January 27, 2015

this part was great! (spoiler?)
After a few weeks, Iwaniec and Friedlander wrote to Katz, “We have completed our study of the paper ‘Bounded Gaps Between Primes’ by Yitang Zhang.” They went on, “The main results are of the first rank. The author has succeeded to prove a landmark theorem in the distribution of prime numbers.” And, “Although we studied the arguments very thoroughly, we found it very difficult to spot even the smallest slip. . . . We are very happy to strongly recommend acceptance of the paper for publication in the Annals.”

Once Zhang heard from Annals, he called his wife in San Jose. “I say, ‘Pay attention to the media and newspapers,’ ” he said. “ ‘You may see my name,’ and she said, ‘Are you drunk?’ ”
also i think i spotted a typo? "Zhang spent a month at the Chis’ [sic]."

the documentary looks fun too btw; here he talks about it as a macarthur fellow and more on polymath8 here :P

oh and re: refereeing...
-Mathematician's anger over his unread 500-page proof [1,2]
-ON THE VERIFICATION OF INTER-UNIVERSAL TEICHMÜLLER THEORY: A PROGRESS REPORT (AS OF DECEMBER 2014)
posted by kliuless at 9:54 AM on January 27, 2015 [3 favorites]

There are an infinite number of pairs of primes no more than 246 apart from each other.

This single sentence is clearer than any of the explanations in the article itself. I was trying to get the point from the article, but for me at least the ruler metaphor and the pigeon metaphor seemed to contradict each other.
posted by Pyrogenesis at 9:58 AM on January 27, 2015 [2 favorites]

Doesn't the FPP directly contradict that statement?

No. The formulation in both you and ilicet's comments are incorrect. Read the previously link... "he announced that he had proved that there are infinitely many pairs of prime numbers separated by no more than 70,000,000" is correct.

That is in fact the beauty of the result. Primes get sparser in the number line as you go higher, but we now know that there will always be pair clusters (no farther apart than 246) no matter how high you go.

Is it simply that 247! + 2 can be factored into 2*(247!/2 + 1)? that seems legit......

Yep. It basically a constructive proof that for any n, you can find a sequence of n numbers that are not prime.
posted by kmz at 9:58 AM on January 27, 2015 [4 favorites]

Gordion Knott, those are all sets of prime numbers which meet some criteria. Just because the criteria was chosen more or less arbitrarily doesn't mean a mathematician can't prove stuff related to it. I mean, most of the problems mathematicians are interested in are arbitrary. That seems to be what a lot of mathematicians like about it. I recall some important mathematician being quoted as saying he was proud that his work served no useful purpose.
posted by Green With You at 9:58 AM on January 27, 2015

Thanks for clarifying, kmz! I confused myself twice over while rereading the article, but I understand now.

I loved a lot of the anecdotes in the profile. "Are you drunk?" is great.
posted by ilicet at 10:04 AM on January 27, 2015

Yes, thanks kmz--that makes more sense.
posted by yoink at 10:12 AM on January 27, 2015

Why was the phrase "bound gaps" used instead of "gap bounds"? Am I missing something here?
posted by Jpfed at 10:13 AM on January 27, 2015

Why was the phrase "bound gaps" used instead of "gap bounds"?

Because the "gap" between any prime and its nearest partner is "bounded" (i.e., has an upper limit), no?

The simplest way of stating this, then, would be that there is a finite gap between any given prime and the next nearest prime, but there is an arbitrarily large gap between that pair of primes and the next prime-pair. Does that about sum it up?
posted by yoink at 10:19 AM on January 27, 2015

If you become a good calculus teacher, a school can become very dependent on you. You’re cheap and reliable, and there’s no reason to fire you. After you’ve done that a couple of years, you can do it on autopilot; you have a lot of free time to think, so long as you’re willing to live modestly.

It seems like anytime a big heady breakthrough like this occurs, it's coming from some Russian recluse or closet genius living outside the academic system. It makes me wonder if the competitive academic environment actively discourages this type of long-term ambition in favor of shorter term prestige.

I know it's more likely that significant breakthroughs happen from within the establishment as well, and that it just doesn't make as interesting a story as the outsider genius. But still, I wonder.
posted by Think_Long at 10:22 AM on January 27, 2015 [3 favorites]

Because the "gap" between any prime and its nearest partner is "bounded" (i.e., has an upper limit), no?

The gap between any prime and its nearest partner is not bounded. Rather, there is a bound on the smallest gap (between consecutive primes) that occurs infinitely often.

The phrase used was not "bounded" gaps, it was "bound" gaps. This phrase seems to imply either gaps bound to something, or gaps between bounds (?). "Gap bounds" seems like a much more appropriate phrase, because we are talking about bounds on gaps.
posted by Jpfed at 10:26 AM on January 27, 2015

It has been known for a very long time that there are an infinite number of prime numbers (numbers divisible only by themselves and 1). We also know that they get less and less common as you go on (and we know by how much they get less common). We also know (and this is pretty easy to show) that you can have arbitrarily large gaps with no primes. Want a trillion consecutive numbers with no primes? Easy. But there seem to be a lot of small gaps, even when you get quite far out. It looks pretty random, to be honest?

But is it? (dun dun DUNNNNN!)

One of the older prime questions is if there are an infinite number of pairs of primes that differ by two. 5 and 7, for example. Or 101 and 103. We don't know (although I think most mathematicians think that this is true). In fact, up until a few years ago, we didn't know if there were an infinite number of pairs primes that differed by any particular finite number. Are there an infinite number of primes that differ by a hundred? A million? (Note: I'm not sure here if it is necessary that there be no other primes in between them). We just didn't know. In fact, is there any number 'n' for which you can say that there are an infinite number of pairs of primes that differ by no more than 'n'? We didn't even know that. For all we knew, all prime gaps might only "appear" a finite number of times.

What Zhang showed is that there are a infinite number of pairs of primes 70000000 apart. There can be bigger gaps between consecutive primes. There can be smaller gaps. But there are infinitely many gaps of this size. And, as it turns out, this number can be reduced quite dramatically (but not yet down to 2).
posted by It's Never Lurgi at 10:28 AM on January 27, 2015 [17 favorites]

Jpfed, while there is some evidence to suggest you are correct in terms of common english words (example), I believe bound == bounded is common usage in mathematical terminology.
posted by grog at 10:30 AM on January 27, 2015

The gap between any prime and its nearest partner is not bounded. Rather, there is a bound on the smallest gap (between consecutive primes) that occurs infinitely often.

Ah, o.k. thanks. I've been very dense about this. The formulation in the FPP just threw me right off.
posted by yoink at 10:33 AM on January 27, 2015

Ah, sorry I worded that badly--but yeah, as I understand it Zhang proved that a finite bound of 70 million, placed at any part of the infinite number line, will contain a pair of primes.

No, but if you move it up the number line, you will eventually find a pair of primes that fit within it. And if you move up again, you'll find another, and another, and so on.
posted by empath at 10:36 AM on January 27, 2015 [1 favorite]

It makes me wonder if the competitive academic environment actively discourages this type of long-term ambition in favor of shorter term prestige.

Yes, many of the quotes from Zhang's colleagues and peers in this piece (whose grudging praise made many of them sound rather awful, at least to me) are basically confessions of this problem and/or unconvincing rationalizations about how it's really mostly okay because Zhang is such an exceptional case. The current academic job system strongly disincentivizes ambitious long-term projects (like this one) that won't also produce a steady stream of easily publishable work along the way to support employment and tenure. It's an acknowledged problem in other fields too — that no one but the already-tenured can afford to take ten years to publish a landmark history or a new dictionary, etc., because they won't have a job by the time a work of that scale is ready to publish —  but it seems like a particular problem in mathematics of this kind, where it could take an indeterminate number of years to reach whatever the crucial breakthrough turns out to be, or you might just never get there.

Zhang, who also calls himself Tom
Her name is Yaling, but she calls herself Helen.

This seems in slightly poor taste to me. I hope The New Yorker comes up with a better house style for people with different names in different languages.

posted by RogerB at 10:57 AM on January 27, 2015 [4 favorites]

The current academic job system strongly disincentivizes ambitious long-term projects (like this one) that won't also produce a steady stream of easily publishable work along the way to support employment and tenure.

The PI for the visual perception lab I worked for bemoaned this. Later, he was denied tenure and he left the university. Unlike mathematics, his project wasn't the sort of work he could do while working at a Subway, so that's that for him.
posted by Jpfed at 11:15 AM on January 27, 2015 [1 favorite]

It was a really fascinating article, but I agree with RogerB, the "calls him/herself" language seemed very weird to me as well. While accurate in a very literal sense, it's language I'd use to describe a con artist or someone who was putting on airs, not someone who had a normal nickname or a separate English name. Something like "goes by" or "uses the name" or "uses the English name" would have read a lot more neutrally to me (e.g. "Her Chinese name is Yaling; in the States, she uses the English name Helen").
posted by en forme de poire at 11:21 AM on January 27, 2015

Gordion Knott, those are all sets of prime numbers which meet some criteria.

Maybe I'm being dense about this, but it seems to me that the holey primes and curved-digit primes aren't sets of prime numbers--they're sets of random, graphical patters that, in mathematics based on the Arabic writing system, are used to refer to those prime numbers. In other words, the squiggly line that signifies an "eight" (8) isn't an eight, but rather the arbitrary symbol chosen by a forgotten linguist in ancient Arabian history to represent that quantity.

This being the case, we can expect holey primes and curved-digit primes to have completely different values based on the writing system--Arabic, Japanese, Sindarin, Klingon, language-I-just-invented-on-this-napkin--that I'm using at the time.

I realize that mathematicians like to talk about arbitrary or useless phenomena, but in this case, they seem to be substituting a random referent for the actual number. Holey and curved-digit primes are not about numbers, they're about semiotics.
posted by Gordion Knott at 11:43 AM on January 27, 2015 [1 favorite]

I am such a tourist in the world of numbers
posted by From Bklyn at 11:43 AM on January 27, 2015

This is nothing compared to my forthcoming blockbuster paper: All Primes Ending with the Number 4.
posted by dances_with_sneetches at 11:46 AM on January 27, 2015

Ah, sorry I worded that badly--but yeah, as I understand it Zhang proved that a finite bound of 70 million, placed at any part of the infinite number line, will contain a pair of primes.

That's not quite my understanding. I believe that the formula

lim inf (pn+1 − pn) < 7 × 107
n→∞

indicates that as you go farther up the number line (n→∞) the limit of the infimum (greatest lower bound) of the distance between primes is less than 7*10**7 (or some smaller number, like 2!). Since you can keep going up the line as far as you want, this means there will always be an infinite number of such pairs. If the bound is 2, then there are an infinite number of twin primes, the original conjecture.

(On preview, what empath said.)
posted by Mental Wimp at 11:50 AM on January 27, 2015

Yeah, but it's semiotics that rigorously defines a set of numbers. Does it matter if it does so arbitrarily? The set of numbers which are primes and which, in modern German, must be spelled with the letter "z" is a rigorously defined set; that it doesn't correspond with the equally rigorously defined set "numbers which primes and which, in modern English, must be spelled with the letter 'z'" doesn't make that any less the case, does it?

I can imagine there might be good reasons (cryptography, say?) where you would want to be able to rigorously define a non-finite set of primes which has no inherent mathematical coherence.
posted by yoink at 11:55 AM on January 27, 2015 [1 favorite]

Maybe I'm being dense about this, but it seems to me that the holey primes and curved-digit primes aren't sets of prime numbers--they're sets of random, graphical patters that, in mathematics based on the Arabic writing system, are used to refer to those prime numbers.

Yes, they're sets of prime numbers that, when written in Arabic numerals, have digits with a particular property. But that doesn't make them not sets of prime numbers, it just makes them possibly silly, not-that-interesting, somewhat arbitrary sets of prime numbers — at least mostly; in my rather dim recollection of number theory I vaguely recall that there sometimes are non-silly reasons why you might be interested in what digits appear in a number's written representation, and a few of the ones listed here seem like they might potentially qualify. But as far as I can tell the main point of bringing them up was just to illustrate in a cute, non-mathematician-accessible way that, you know, people are really into prime numbers. ("Fetishistic" about them, speaking of the abuse of technical terms.)
posted by RogerB at 11:56 AM on January 27, 2015 [1 favorite]

Can I just say that I think it's in our interest, as a species, to make it so that people like Zhang shouldn't have to scratch out a living doing menial things while furtively stealing little scraps of time to do the thing they love? (I write this from work, where I just spent my lunch hour working on writing a novel.). That that's really kind of the central subtext of the article - why should this guy have been working at a sandwich shop?
posted by newdaddy at 12:06 PM on January 27, 2015 [2 favorites]

Maybe I'm being dense about this, but it seems to me that the holey primes and curved-digit primes aren't sets of prime numbers--they're sets of random, graphical patters that, in mathematics based on the Arabic writing system, are used to refer to those prime numbers.

Oh no. The shape of a digit is determined by the number of factors it has. Prime digits do not divide the plane (i.e.; they have no loops). Digits with two prime factors divide the plane into least two parts: they have one loop.(*) The only digit to have three prime factors (8) divides the plane into three parts.

Surely it's worth seeing whether and how this relationship continues as we proceed through multi-digit primes.

(*) And now you know how the digit "4" is correctly constructed.
posted by Joe in Australia at 12:37 PM on January 27, 2015 [4 favorites]

I think Gordion Knott's point is that several of these categories have nothing to do with the number itself. Like, aliens who had no contact with Earth could never determine whether a number is a member of the set of curved-digit primes the way they could determine whether a number is a Mersenne prime or is a prime that's a palindrome when rendered in base 10.

Joe in Australia, zero does not fulfill the criteria you describe but appears to be included in the definitions of both holey and curved-digit primes.
posted by XMLicious at 12:50 PM on January 27, 2015

kmz: Remember that this is a weaker form of the Twin Primes conjecture, which says that there's an infinite number of primes two away from each other. Obviously there's not always two primes in any set of 3 consecutive integers.
Ah, yes; how foolish of me to forget...
posted by IAmBroom at 1:01 PM on January 27, 2015

kmz: Doesn't the FPP directly contradict that statement?

No. The formulation in both you and ilicet's comments are incorrect.
The formulations mentioned are incorrect, so they are not supported by the FPP, and in fact are contradicted by it... so by "No" you mean "Yes"?
posted by IAmBroom at 1:07 PM on January 27, 2015

Mental Wimp: (or some smaller number, like 2!)
Heh. Is that an intentional math joke? (2! is indeed a smaller number.)
posted by IAmBroom at 1:09 PM on January 27, 2015 [1 favorite]

I certainly wasn't saying that holey primes are as interesting as palindromic prime numbers. An alien, if it even uses positional numerals the way we do, might have considered similar things in their own number system though. I think I'd like to know that there are aliens goofing off with numbers. If an alien said to me "These numbers are interesting to us because to us they look like x!" or something like, that I'd kinda geek out. If they were more interested in those than other stuff I might look askance at their culture though.
posted by Green With You at 1:11 PM on January 27, 2015

Heh. Is that an intentional math joke? (2! is indeed a smaller number.)

Read it either way. Knock yourself out!
posted by Mental Wimp at 1:13 PM on January 27, 2015 [1 favorite]

The formulations mentioned are incorrect, so they are not supported by the FPP, and in fact are contradicted by it... so by "No" you mean "Yes"?

Damnit Jim, I was math major, not an English perfect speaking thingy. (Or an epistemologist of whether that's a logic or language issue.)
posted by kmz at 1:18 PM on January 27, 2015

You can do fairly interesting things with somewhat arbitrarily defined sets, for example, you can define numbers as representing logical symbols and operators, and create "the set of numbers that correspond to logical statements", which are arbitrary in two ways-- the particular mapping of numbers to symbols, and the particular notation for formal logic-- but you can proove some fairly significant results in that way.
posted by empath at 1:27 PM on January 27, 2015

also i think i spotted a typo? "Zhang spent a month at the Chis’ [sic]."

Nope. The couple's surname is Chi, together they are the Chis, so their house is the Chis' house. He spent a month at the Chis' [house].
posted by rory at 1:36 PM on January 27, 2015 [1 favorite]

got it, tks! nyër ftw :P
posted by kliuless at 1:49 PM on January 27, 2015 [2 favorites]

This article has some good stuff, but I was struck by how unfamiliar it seemed to be with academic life. Like, he's surprised that Zhang just has books and a desk and a chair in his office? Or that he just works on math all the time to the exclusion of other things, or that he's able to teach his friend's kid calculus without a syllabus, just from memory?... Like, of course. Have you met academics, dude? And the phrase "algebraic geometrist" (should be "geometer")... and the description of how peer review works (which is good to include but struck me as being maybe excessively 'this is obscure insider stuff'ish?)... I don't know, the whole thing had a weird "exploring a strange unknown world" feeling, when surely a New Yorker writer knows a bunch of academics?

And the description of the math is confusing, I think. E.g., the phrase "on infinitely many occasions" is just odd when what it should be is "at infinitely many positions on the number line" or similar. Cf the confusion we started this thread with.

Anyway, enough grumbling, I was glad to read this. I'm sure Zhang is a tough subject for this kind of profile, since he's so reserved and uninterested in publicity, and it was really interesting to read a bunch of the background of his life and hear the referees' notes. I definitely don't think the mathematicians quoted were trying to damn him with faint praise -- those quotes seem pretty effusive, for mathematicians.
posted by LobsterMitten at 2:11 PM on January 27, 2015 [2 favorites]

In the most basic terms:

Primes are numbers that can't be evenly divided by anything else: 2, 3, 5, 7, 11. If you experiment, you find that primes are rarer as your numbers get bigger.

The first great non-obvious discovery about primes is the Prime Number Theorem - which says that the chance of a number being prime is tends to be inversely proportional to its number of digits - in other words, a number that's twice as long has (roughly) half the chance of being prime.

A bit more experimentation shows you lots of primes that are just two positions apart: 5/7, 11/13, 17/19, ..., 101/103, ... These are called twin primes.

Since the primes are always getting further apart on average, in the long run you'd expect to never see twin primes at all - in other words, that there are only finitely many twin primes.

But experimentation seems to show you do find twin primes no matter how large your numbers are - but no one was able to understand why that might be, or even if it were true in general.

And there it basically sat for a hundred years, with only a little action around the edges.

Finally, Yiting Zhang (and what an amazing romantic story it is, and what persistence) came out of nowhere and said, "I can't show that there are infinitely many pairs of primes that are 2 places apart - but I can show that there are infinitely many pairs of primes that are less than 70,000,000 places apart!"

This doesn't sound like much - but before then, there was no reason to suppose that primes didn't go off into the dark huge space of very large numbers and end up isolated, far apart.

Zhang showed we have an explicit bound to this process - that no matter how large your numbers you're looking at get, you'll always eventually run into two primes that are closer together than some number, GAP.

Now, GAP = 70,000,000 in the initial paper but that's the sort of thing that people can get their teeth into, so with a bunch of distributed work, they've gotten it down to 246. True success would be if they get GAP = 2 - but that's seeming out of reach from the techniques so far...
posted by lupus_yonderboy at 2:12 PM on January 27, 2015 [6 favorites]

I'd say the first non-obvious discovery (from Euclid) about primes is that there are infinitely many of them.

To prove it, imagine that you have a list of all the prime numbers. Multiply them all together, then add 1. You now have a number which is not divisible by any other prime, making it a new prime.
posted by empath at 2:22 PM on January 27, 2015 [2 favorites]

This is nothing compared to my forthcoming blockbuster paper: All Primes Ending with the Number 4.

My latest conjecture posits that there is an infinite number of even primes.
posted by Mr.Encyclopedia at 2:42 PM on January 27, 2015 [2 favorites]

> I'd say the first non-obvious discovery (from Euclid) about primes is that there are infinitely many of them.

Heh, yes, I suppose it depends on your definition of obvious ("After an hour, he looked up from the paper. "I was right! It is obvious."")

There are a bunch of others, like the Unique Factorization Theorem, but the Prime Number Theorem is the first one where you can't give a one-line explanation like you gave. It's not obvious at all, and it takes you weeks of work to prove it in a number theory class.
posted by lupus_yonderboy at 2:57 PM on January 27, 2015 [1 favorite]

To prove it, imagine that you have a list of all the prime numbers. Multiply them all together, then add 1. You now have a number which is not divisible by any other prime, making it a new prime.

not quite - it is either prime, or its prime factors are not on your original list, contradicting the assumption.

example: 2 x 3 x 5 x 7 x 11 x 13 + 1 = 30031 = 59 x 509
posted by mr vino at 3:55 PM on January 27, 2015 [6 favorites]

I have a Chinese friend, an engineer, who speaks very much the same way and who is called Joe though that is not his name. I kept hearing and seeing him while reading the profile.
posted by Peach at 4:37 PM on January 27, 2015

> ON THE VERIFICATION OF INTER-UNIVERSAL TEICHMÜLLER THEORY: A PROGRESS REPORT (AS OF DECEMBER 2014)

Pure Borges.
posted by benito.strauss at 9:48 PM on January 27, 2015 [1 favorite]

Gordion Knott, you've got a point, but I'm guessing you're a philosopher, because you're stating it in terms of "semiotics" versus "actual number". A model theorist (i.e. an obscure type of mathematician) would make the distinction in terms of what language you use, where we can think of "language" as the operations and constants used.

So "twin primes" can be defined using (more or less) just 0, 1, plus, minus, and times, and some set of logic, (I'm not sure if you need second order or can get away with just first order). Let's call this language LTP (language for twin primes).

But the holey primes (and the other similar categories) require you to talk about "the digits of x in base 10". While you could probably use LTP to define a function

dig( x, b, k) = the kth digit of the number x in base b,

the "holey primes" require you to write statements where this function is evaluated only at b=10. [By the way, if you want to stay in LTP, 10 has to be considered a shorthand for 1+1+1+1+1+1+1+1+1+1.] And the conditions for being "holey" includes testing whether this dig() function is equal to 1+1+1+1, or 1+1+1+1+1+1, 1+1+1+1+1+1+1+1, or 1+1+1+1+1+1+1+1+1. What you end up with is a whole bunch of (arbitrary) constants showing up in your definitions, You've still strictly defined a set of numbers, but the more arbitrary values you've got, the less general or basic your result feels.

Mind you, in this view the "2" in the definition of twin primes would also be considered an arbitrary constant, but it's the only one required, and the number 2 is generally viewed as less arbitrary than the number 10. </math humor>
posted by benito.strauss at 10:27 PM on January 27, 2015

Why was the phrase "bound gaps" used instead of "gap bounds"? Am I missing something here?

I suspect this is an error by the New Yorker. The title of Zhang's paper refers to "bounded gaps", and "bounded" is absolutely the standard terminology all across mathematics, and especially common in analysis. There is a sequence of pairs of consecutive primes, and the gaps between them are bounded. I don't think I've ever heard "bound" used to mean "bounded" previously.
posted by stebulus at 2:59 AM on January 28, 2015 [2 favorites]

I have defined a new category called "purple primes", which when represented as a hexadecimal number and divided into a series of six-digit groups, each of which are interpreted as a 24-bit RGB color within the color gamut of late 20th century electronics, if those six-digit groups are averaged by CIEDE2000 color distance the resulting value appears as a shade of purple in human vision. Also, when spoken aloud in Medieval Crimean Gothic the number must fit into iambic pentameter.
posted by XMLicious at 6:04 AM on January 28, 2015 [1 favorite]

not quite - it is either prime, or its prime factors are not on your original list, contradicting the assumption.

example: 2 x 3 x 5 x 7 x 11 x 13 + 1 = 30031 = 59 x 509

Well, yes, then you didn't start with a list of all the primes.
posted by empath at 9:32 AM on January 28, 2015 [1 favorite]

Hmm.....

(Tries to think about a ring extending ℤ where "product of all finite primes" makes sense. Gets headache.)
posted by benito.strauss at 12:00 PM on January 28, 2015 [1 favorite]

a ring extending ℤ where "product of all finite primes" makes sense

Well, there's ℚ. (Everything's a unit, so the product is empty.)
posted by stebulus at 5:34 PM on January 28, 2015

You're no fun anymore.
posted by benito.strauss at 5:56 PM on January 28, 2015

Prime Gap Grows After Decades-Long Lull - "Mathematicians have made the first major advance in 76 years in understanding how far apart prime numbers can stray."
Dozens of mathematicians then put their heads together to improve on Zhang’s 70 million bound, bringing it down to 246 — within striking range of the celebrated twin primes conjecture, which posits that there are infinitely many pairs of primes that differ by only 2.

Now, mathematicians have made the first substantial progress in 76 years on the reverse question: How far apart can consecutive primes be? The average spacing between primes approaches infinity as you travel up the number line, but in any finite list of numbers, the biggest prime gap could be much larger than the average. No one has been able to establish how large these gaps can be.
posted by kliuless at 8:32 PM on January 29, 2015

Sure, sure, he proved his theorem, but did he do it in haiku?
posted by Mental Wimp at 2:17 PM on February 6, 2015

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