Quite a day for analytic number theory

May 13, 2013 8:26 PM

This afternoon, Yitang Zhang of the University of New Hampshire gave a special seminar at Harvard, in which he announced that he had proved that there are infinitely many pairs of prime numbers separated by no more than 70,000,000, a result differing only by a constant factor from the venerable twin prime conjecture. Dan Goldston, who together with Yildirim and Pintz made the last major advance on prime gaps, said, ""I was doubtful I would ever live to see this result." Not enough excitement for one day? Harald Helfgott has just posted to the arXiv a proof of the ternary Goldbach conjecture: every odd number is the sum of three primes.
posted by escabeche (54 comments total)

Dammit. I took the over.
posted by Etrigan at 8:32 PM on May 13, 2013 [2 favorites]

differing only by a constant factor from the venerable twin prime conjecture

Mathematically accurate dry understatement is the best kind of dry understatement.
posted by figurant at 8:32 PM on May 13, 2013 [17 favorites]

Goldston, who was sent a copy of the paper, says he and the other researchers who have seen it "are feeling pretty good" about it. "Nothing is obviously wrong," he says.

Oh, mathematicians.
posted by deathpanels at 8:34 PM on May 13, 2013 [6 favorites]

I love this!
posted by ILuvMath at 8:34 PM on May 13, 2013 [3 favorites]

This is true holycrap news. These simple-to-state and simple-to-understand conjectures are both centuries old.
posted by hexatron at 8:35 PM on May 13, 2013

I seriously just did a fist pump in the air over the ternary Goldbach conjecture. That made me VERY happy.
posted by strixus at 8:39 PM on May 13, 2013

The ternary Goldbach conjecture, or three-primes problem, asserts that every odd integer N greater than 5 is the sum of three primes. The present paper proves this conjecture.

Not the sort of abstract you get in many biology papers.
posted by straight at 9:01 PM on May 13, 2013 [12 favorites]

t's a result only a mathematician could love. 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's only [a factor of] 35 million away" from the target, quips Dan Goldston, an analytic number theorist at San Jose State University in California who was not involved in the work. "Every step down is a step towards the ultimate answer."

This reminds me of Graham's number, which was also devised to serve as an upper bound on the solution to a mathematical problem. But it is so huge (many times larger than googolplex) that special notation has to be used to write it. Despite this enormous upper bound, Graham and Rothschild conjectured that the answer was between 6 and 10!<sup>1

1</sup> It has since been proven to be at least 13.
posted by grouse at 9:08 PM on May 13, 2013 [4 favorites]

It's kind of amusing how both of these huge achievements are proofs of "weaker" conjectures that would be trivially true if the "strong" versions (which both seem very likely to be true) could be proved.

If it's true that every integer is the sum of 2 primes, then of course every odd integer is the sum of three primes. And if there are an infinite number of primes separated by 2, then "at most 70,000,000" seems pretty funny.
posted by straight at 9:15 PM on May 13, 2013

And it really gives me a nice sense of my vast depths of ignorance about math trying to imagine what sort of proof could possibly yield the "at most 70,000,000" result.
posted by straight at 9:19 PM on May 13, 2013 [3 favorites]

I don't know if anyone noticed this, but the proof of the ternary Goldbach conjecture implies that every even number greater than twelve can be expressed as the result of no more than six primes.

I suppose you would have to prove the cases of 2, 4, 6, 8, 10, and 12 via computation.
posted by Joe in Australia at 9:25 PM on May 13, 2013 [2 favorites]

Neat findings, but how do you monetize them?

/Canada
posted by Sys Rq at 9:27 PM on May 13, 2013 [33 favorites]

a proof of the ternary Goldbach conjecture

Meh, I'll be impressed when they prove Strong Goldbach. I don't expect that to ever happen. Just looking quickly at the paper, I was even less impressed:

"The main theorem has been checked deterministically by computer for all n < 10^30.. Numerical verifications of the binary Goldbach conjecture for small n were published already in the late nineteenth century;"

A brute force proof by computer, how inelegant.

I was surprised to have the Goldbach Conjecture come up in my job, alas under circumstances I can't reveal due to a nondisclosure agreement. I was sitting in a room full of mathematicians, of which I am the least qualified. One of them is working a prime number problem on the whiteboard in front of everyone, and I said hey that formula looks familiar, don't you recognize it? That's a conjecture.. I can't remember which.. (quick Mathworld & wikipedia crosscheck) Oh it's the Goldbach Conjecture. Some of them were stunned that they missed it. I was more stunned when some of the mathematicians said they never heard of it.
posted by charlie don't surf at 9:29 PM on May 13, 2013 [2 favorites]

I don't know if anyone noticed this, but the proof of the ternary Goldbach conjecture implies that every even number greater than twelve can be expressed as the result of no more than six primes.

For what it's worth, this fact was proved by Olivier Ramare in 1995.
posted by escabeche at 9:30 PM on May 13, 2013

I was surprised to have the Goldbach Conjecture come up in my job, alas under circumstances I can't reveal due to a nondisclosure agreement.

Hey, buddy; we've all been there.
posted by yoink at 9:50 PM on May 13, 2013 [18 favorites]

I don't know if anyone noticed this, but the proof of the ternary Goldbach conjecture implies that every even number greater than twelve can be expressed as the result of no more than six primes.

Four, not six. An even number can be expressed as the sum of an odd number and a prime.
posted by kickingtheground at 9:56 PM on May 13, 2013 [7 favorites]

I must not try to make jokes in maths threads.
I must not try to make jokes in maths threads.
I must not try to make jokes in maths threads.
...
posted by Joe in Australia at 9:59 PM on May 13, 2013

Meh, I'll be impressed when they prove Strong Goldbach.

Hipster mathematicians ???
posted by Podkayne of Pasadena at 10:10 PM on May 13, 2013 [10 favorites]

I immediately thought "1 is odd!"

But the conjecture is actually "every odd number greater than 7 can be expressed as the sum of three odd primes."
posted by Foosnark at 10:14 PM on May 13, 2013 [1 favorite]

"The main theorem has been checked deterministically by computer for all n < 10^30.. Numerical verifications of the binary Goldbach conjecture for small n were published already in the late nineteenth century;"

A brute force proof by computer, how inelegant.

The main achievement in the paper was proving it for all n > 10^30 so that they could mechanically verify the rest.
posted by empath at 10:16 PM on May 13, 2013 [15 favorites]

Meh, I'll be impressed when they prove Strong Goldbach.

Yeah, Wiles! Don't come back until you've proved the full Taniyama–Shimura conjecture!
posted by figurant at 10:17 PM on May 13, 2013 [2 favorites]

TNAANT, but quite pleased to see this!
posted by kengraham at 10:18 PM on May 13, 2013

Meh, I'll be impressed when they prove Strong Goldbach

That man is the worst nuisance on the beach!
posted by benzenedream at 10:23 PM on May 13, 2013 [3 favorites]

The main achievement in the paper was proving it for all n > 10^30 so that they could mechanically verify the rest.

OK that sounds more like a proof. Obviously I lose interest almost instantly when I see brute force computation.
posted by charlie don't surf at 10:30 PM on May 13, 2013

This will certainly make for good material on the I Fucking Love Math page.
posted by Apocryphon at 10:41 PM on May 13, 2013

It's kind of strange how the sorts of trivial (and probably useless for any practical purpose, if proven) conjectures that a smart 10th grader might scribble down on his notebook are so hard to prove.
posted by empath at 11:12 PM on May 13, 2013 [1 favorite]

These are two of the four Landau's problems. Besides there two there are the Legendre Conjecture (that there is always a prime between successive squares) and the conjecture that there are infinitely many primes of the form n²+1.

They are so named because Landau mentioned them in connection with each other in 1912.
posted by tykky at 11:22 PM on May 13, 2013

Obviously I lose interest almost instantly when I see brute force computation.

I know how you feel. Seeing certain features of some things make me reflexively ignore them too.
posted by benito.strauss at 12:18 AM on May 14, 2013 [3 favorites]

It's kind of strange how the sorts of trivial (and probably useless for any practical purpose, if proven) conjectures that a smart 10th grader might scribble down on his notebook are so hard to prove.

Well, it's more like "All the trivial conjectures that couldn't be proven with the math techniques of the 1800s by some of the greatest mathematicians who ever lived are so hard to prove." All the easy stuff has been taken.

And about brute force computation: Without it we don't have the Four Color Theorem or the classification of the finite groups, so put me in Camp Brute Force Computation. </bandname>
posted by Elementary Penguin at 12:23 AM on May 14, 2013 [1 favorite]

I'm kind of curious about how Zhang, the prime-pairs paper author looks middle-aged in his photo, and the linked article claims he discovered the key insight of his proof last year. One bit of conventional wisdom I often hear about mathematicians at this level is that they make any crucial discoveries very early in their career. Thus I wonder, is this a false generalization about mathematicians, or is Zhang just an exception? (Or is his result in fact building on his earlier work?)
posted by dendrochronologizer at 12:25 AM on May 14, 2013

I think it's more that it's less resource-intensive to make a breakthrough in math. You can 'just' hyperfocus on one problem and write a proof and it's either right or wrong. You don't need to do years of expensive research, get grants, establish a reputation, build a lab and so on. So people break through younger. But that doesn't mean that they stop working. Theoretical physics can be the same way.
posted by empath at 12:39 AM on May 14, 2013

It's kind of strange how the sorts of trivial (and probably useless for any practical purpose, if proven) conjectures that a smart 10th grader might scribble down on his notebook are so hard to prove.

This is a comment I've heard a few times about Number Theory (which is what these two problems are), in particular anything involving Primes. Alone among current research areas, it is comprehensible and interesting to a young math enthusiast. Cutting-edge problems, and sometimes even their solutions, can be understood by someone willing to dig in directly. Most other fields of math require many years of pre-requisites (often in areas that seem not to be relevant) before you get to the point where one can even begin to study them.
posted by Harvey Kilobit at 1:09 AM on May 14, 2013 [2 favorites]

I'd day Roger Apéry and his proof of ζ(3)'s irrationality is one good example of an older mathematician discovering a new and unexpected result. According to Wikipedia, he was over 60 years old when he published the proof.

It is probably true that most new breakthroughs are made by younger mathematicians, though. But we shouldn't allow this as an excuse for older mathematicians to slack, should we??? :)
posted by tykky at 1:09 AM on May 14, 2013

Strong Goldbach

He allows every prime number to draw blood... just once.
posted by Slap*Happy at 4:27 AM on May 14, 2013 [3 favorites]

I looked at the title, immediately went "what about 1 3 and 5, this is bullshit", clicked the link, and voila - "odd integer N greater than 5 is the sum of three primes"
posted by Veritron at 5:17 AM on May 14, 2013

One bit of conventional wisdom I often hear about mathematicians at this level is that they make any crucial discoveries very early in their career.

I don't think this is so true anymore; I wrote about this in Slate a while back. Helfgott is 35, not exactly an ingenue. Of course there are lots of twentysomethings doing amazing work, too!
posted by escabeche at 5:44 AM on May 14, 2013 [1 favorite]

...under circumstances I can't reveal due to a nondisclosure agreement.

This is our generation's "which this margin is too narrow to contain."
posted by Riki tiki at 6:03 AM on May 14, 2013 [6 favorites]

Meh, I'll be impressed when they prove Strong Goldbach.

OOH OOH MY TURN

Strong Goldbach was, by far, my least favorite Homestar Runner character.
posted by DoctorFedora at 6:14 AM on May 14, 2013 [7 favorites]

This reminds me of Graham's number, which was also devised to serve as an upper bound on the solution to a mathematical problem. But it is so huge (many times larger than googolplex) that special notation has to be used to write it.

Not to derail too much, but if you want to try and imagine how large Graham's number is, consider this. If each digit of Graham's number occupied one Planck volume, the observable universe would not be large enough to contain it. Compare this with a googolplex. If we do the same thing (each digit occupying one Planck volume), we would only need 0.0000422419 m³ of space for it (about the amount of space an ounce of water takes up).

So, when you say that Graham's number is "many times larger than a googolplex", you ain't kiddin'.
posted by King Bee at 6:15 AM on May 14, 2013 [8 favorites]

And about brute force computation: Without it we don't have the Four Color Theorem

Yeah, to me, that's the exact point where everything went off the rails. Instead of proving a theorem, you have to prove the computation correctly represents an exhaustive set of solutions of the theorem. IMHO it is possible to have one too many levels of abstraction.
posted by charlie don't surf at 7:55 AM on May 14, 2013

Strong Goldbach was, by far, my least favorite Homestar Runner character.

Oh, come on. I love all those alternate-reality Strong Bads. Old-Timey Strong Bad, Manga Strong Bad, Keyboard-Head Strong Bad, Sinistrong Bad, Genius Mathematician Strong Bad.
posted by straight at 8:30 AM on May 14, 2013

This "young mathematician" thing has more or less stopped being a thing. Wiles was in his 40s when he proved Fermat's Last "Theorem"; Perelman a similar age when he proved the Pointcaré conjecture.

The reasons are quite simple: for most fields of mathematical endeavor you need to be extremely advanced - in practice, a post-doc - before you can even start to work on the burning issues of our day; and proofs are much, much, much longer than they ever were.

It's kind of a shame, and it's made me glad that I didn't go on in mathematics. Earlier proofs just weren't that bad - long proofs I remember from school include the fundamental theorem of Galois theory, Gödel's Incompleteness Theorem(*), the Fundamental Theorem of Integral Calculus (basically, "integration is anti-differentiation"), but these were at least two orders of magnitude smaller than what we're seeing today - and they just aren't that bad, I could explain any of them to you in 15 minutes even if you had a very limited math background.

And there were still some interesting problems with short proofs - the proof of the undecidability of Hilbert's Tenth Problem had been boiled down to a 20-page proof when I was an undergraduate, and an extremely understandable 20 pages. I remember reading the thing when I was on mushrooms one night, and then coming in the next day and giving a little impromptu talk to a couple of friends as to how it worked (and I could still tell you 30 years later how it basically worked if you asked me).

But the proofs we're see today... oof. I honestly don't understand how anyone can wrap their head about these proofs, frankly. And I'm starting to find them less convincing as a result - if only a tiny number of people are capable of even evaluating them, I really am less sure that they are correct. Many of the famous conjectures did have incorrect "proofs" in the past - the four-color problem had a proof that people considered correct for over a decade until someone found a hole in it, and that was in the nineteenth century, at a time when any mathematician in any branch could probably read any paper in any other branch with a bit of prep.

Don't get me wrong, these people are ultra-smart, but IMHO it would be really easy to say something plausible but that "does not follow" in one step in a 1000 page proof, and have no one else notice it for a long time - particularly if there were very few eyes on that paper to start with.

(* - interestingly enough, he also did a Completeness Theorem, which is basically unrelated but also extremely useful, and is the basis of whole fields of mathematics like non-standard analysis and model theory.)
posted by lupus_yonderboy at 9:33 AM on May 14, 2013 [6 favorites]

@straight, on the subject of pithy abstracts, you may appreciate this famous paper; Title: The critical probability of bond percolation on the square lattice equals 1/2; Abstract: We prove the statement in the title of the paper.

Actually, searching for "we prove the result stated in the title" gets "about 8,380 results" (but actually only 78).
posted by Fat Charlie the Archangel at 10:15 AM on May 14, 2013

Along those lines, there was also a recent paper in the field of self-avoiding walks titled, "The connective constant of the honeycomb lattice equals sqrt(2 + sqrt(2))."

There's a whole thread on MathOverflow for "memorable titles." Most of them are silly jokes (like "Ramanujan's associations with radicals in India") but a few are the ones that just state the whole result in the title, like "M_22 is of general type" and "Primes is in P."
posted by Aquinas at 1:45 PM on May 14, 2013

I presume you meant this link, Aquinas. And now I'm sitting at my desk trying to quake with laughter as silently as possible to paper titles like You Could Have Invented Spectral Sequences by Timothy Y. Chow
posted by figurant at 2:04 PM on May 14, 2013

Finding composite order ordinary elliptic curves using the Cocks-Pinch method, by D. Boneh, K. Rubin and A. Silverberg.

Oh my god, I'm twelve.
posted by figurant at 2:07 PM on May 14, 2013 [2 favorites]

figurant: Hairy Ball Theorem

Now you're ten!
posted by Elementary Penguin at 2:20 PM on May 14, 2013 [1 favorite]

Whoops. Yep, that's the one I meant.

I stumbled across this paper a while back too. While the title "Large embedded balls and Heegard genus in negative curvature" is unremarkable, the introduction has this line towards the end:

A proper subset of the authors wish to subtitle this paper "Big balls imply big genus."
posted by Aquinas at 2:46 PM on May 14, 2013 [4 favorites]

Hairy Ball Theorem
From Wikipedia, the free encyclopedia

"Hairy balls" redirects here. For the mayor of Fort Wayne, see Harry Baals.
posted by figurant at 3:00 PM on May 14, 2013 [5 favorites]

When it comes to large proofs and the credence mathematicians lend them, see the first Related Post, Proof and Community Standards. As an outsider it's very interesting to me the difference in reception of the two purported results: very cautious in the case of the ABC result and relatively optimistic in the case of this story.
posted by jepler at 3:04 PM on May 14, 2013

Yes, but will the writers of Futurama be able to incorporate this into an episode plot?
posted by radwolf76 at 3:35 PM on May 14, 2013

I wrote a piece about Zhang's proof in Slate.
posted by escabeche at 5:55 AM on May 24, 2013 [2 favorites]

Escabeche, Ulam's spiral certainly makes prime numbers look as though they follow a structured distribution. How can this be reconciled with their apparent randomness? Is it just an artefact caused by a regular distribution of composite numbers, which forces the primes into the gaps? And if you took that far enough, wouldn't you effectively have a prime-generating function anyway?
posted by Joe in Australia at 12:52 PM on May 24, 2013

I want to rephrase the above: perhaps Ulam's spiral and other similar depictions just tend to generate composite numbers in positions that are not diagonal lines, and the primes are consequently are more likely to lie on diagonals. But a perfect spiral or combination of spirals would generate all composite and only composite numbers, and would effectively be a formula for generating primes.
posted by Joe in Australia at 1:54 PM on May 24, 2013