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Unheralded Mathematician Bridges the Prime Gap
- chime 13y ago> Among large numbers, the expected gap between prime numbers is approximately 2.3 times the number of digits; so, for example, among 100-digit numbers, the expected gap between primes is about 230. > no matter how sparse the primes become — you will keep finding prime pairs that differ by less than 70 million. Does this mean even for numbers longer than 70m/2.3 = 31m digits, there is bound to be at least one prime every 70m numbers or so?
- jckt 13y agoEdit: Admittedly only skimmed the article, let alone the paper. Obviously there are an infinite number of primes and we've proved that ages ago. Do not read this comment, read the replies if you want a more accurate answer. Leaving it here for posterity. Yup, at least according to the article: "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. [...] [N]o matter how sparse the primes become — you will keep finding prime pairs that differ by less than 70 million." So he proved that it there is some number N, not necessarily 70mil but below 70mil, is the "gap" between primes. -------- This is an amazing development. On the other hand, we are also getting close to nailing the odd Goldbach Conjecture. I think it was just last year that Tao proved, without the Riemann Hypothesis, that every odd number N > 1 can be expressed at by the sum at most 5 primes. Wonderful to witness such great leaps in maths during one's own lifetime.
- rlanday 13y agoNo, that’s not what the paper shows. The paper claims that there exists arbitrarily large primes prime pairs with a gap of less than 70 million, not that all primes are part of a prime pair. Your misinterpretation is equivalent to intepreting the twin prime conjecture to mean that all odd numbers are prime.
- heretoo 13y agoNo, this isn't right. He showed that there are infinitely many twin primes differing by 70 million from each. However, it could be that the next such twin prime, has its lower prime number more than 70million numbers further along. What I mean is, there is allowed to be a gap larger than 70million with no primes in it, then all of a sudden, twin primes. But those twin primes with a gap less than 70million between them are guaranteed. Basically, if you separate all adjacent pairs of primes, with no primes in between, you can separate this into two groups of those adjacent pairs with a gap less or equal to 70mil, or greater than 70mil. The first group has infinitely many members according to the proof. And the second group is probably not empty.
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- ky3 13y agoAnd the second group is probably not empty. We know for sure it's non-empty by simple factorial arguments ITT or by appeal to the prime number theorem.
- impendia 13y agoHarald Helfgott would have a bone to pick with "getting close". :) http://arxiv.org/abs/1305.2897 http://arxiv.org/abs/1305.2897
- jckt 13y agoNot bad for the first half of 2013!
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- CJefferson 13y agoNo, there can be huge gaps with no primes. after n!(n factorial), it is easy to show there are no primes until at least n!+n+1. What this proof shows is you never get to a point where the gap is always bigger than 70 million, forevermore. This is a big step towards the "twin primes conjecture" which claims there are an infinite number of pairs of primes only 2 apart.
- smegel 13y agoI don't get conjectures. Does it mean they are widely believed to be true, just unproven? If so, why, where does this confidence come from? If not, what makes them interesting in the first place?
- kruhft 13y agoThe definition of conjecture says they are basically 'guesswork': http://www.thefreedictionary.com/conjecture http://www.thefreedictionary.com/conjecture. Someone will say 'I think that there are no values of x,y and z can solve x^n+y^n=z^n for n other than 2' but doesn't prove it. This can be due to lack of time, space, or a supreme insight on the part of the conjecturer[1]. The conjecturer has said some important and correct things in the past, so many people will take this experience and look at the conjecture and say 'this is probably right because this guy was right before and we see no errors with this conjecture so we will use it in as a theorem', which is to say they will use it without proof because it makes sense. Then, sometimes 100's of years later, somebody does prove (or disproves) the theorem and get's some press, and for some very important conjectures, prizes. If the conjecture is disproven, then all the math based on it falls down. Conjectures interesting because they are puzzles left by people that they can't answer but they want them to be true, so it is up to future generations with their advanced minds, tools and insights to prove them, leading to progress in the field with the advancements of techniques use to prove these conjectures and make them actual theorems. [1] http://en.wikipedia.org/wiki/Fermat%27s_Last_Theorem http://en.wikipedia.org/wiki/Fermat%27s_Last_Theorem
- ky3 13y agoNot all guesswork is dignified by the term conjecture though. Part of math school is to learn making and breaking guesses about things mathematical. As noted elsewhere ITT, a conjecture becomes one when there's a community-acknowledged aura of moral rightness about it. So at the very least, a bunch of people have tried to disprove it and failed.
- lmm 13y agoNo; trivially there will be stretches of length >70m where there are no primes, and this is true for any length of "desert" you care to specify (e.g. the sequence of 70m numbers starting at (70m+1)! are all composite; so is the one starting at (70m+2)!)). What he's proven is that there's never a point after which every prime is more than 70m away from the last one. Even as the average distance between one prime and the next increases towards infinity, you'll always find occasional pairs that are within 70m of each other; they'll get rarer and rarer as the numbers get larger, but you'll never get to the last such pair.
- impendia 13y agoNo. The average size of a prime gap around numbers of size N is ln(N). http://en.wikipedia.org/wiki/Prime_number_theorem http://en.wikipedia.org/wiki/Prime_number_theorem A rough analogy of this prime gaps work is the following. Imagine that you throw infinitely many darts at a dartboard. Before you throw, draw a bullseye around the center, as small as you want. Then you will miss plenty often, but infinitely many of the darts will hit the bullseye.
- tzs 13y agoNo, it doesn't mean there is a prime EVERY 70m or so. It means that as you go on examining pairs of consecutive primes, you will keep finding more and more where the gap is 70m or less. You will find many more where the gap is more than 70m. The average gap will be increasing. The average gap at N is approximately log(N). (That's the natural logarithm, not the common logarithm). That's where that 2.3 you quoted comes from. If you are looking at numbers with d decimal digits, they are around 10^d and so the average gap is approximately the log of that, log(10^d) is d log(10), and log(10) is approximately 2.3. Hence, the average gap for d digit numbers is approximately 2.3 x d.
- ky3 13y agoOne way of telling the story is this: as we keep traversing ever larger numbers, we can always find a pair of primes whose distance is less than 70 million. The twin primes conjecture holy grail is this: no matter how big we go, we can always find a pair of primes whose distance is the minimal possible, i.e. 2. So as sparse as these pairs become in the limit according to the prime number theorem, the likelihood of finding them remains always positive.
- Xcelerate 13y agoSo, there's two (slightly off topic) questions this article raises that I've had for a while. Perhaps someone who's been in academia longer than I have can enlighten me. First, the article talks about experts within the field. I've been a graduate student for about a year now, and within my field of research (molecular dynamics), I still have no idea who the "experts" are. I see the same names frequently pop up on papers I read -- are these the experts? I don't know of any online forums where people discuss MD; if they're out there, then they must be a secret. I'm sure I can't just email these professors and say "hey, want to chat?" So where does this group collaboration and unanimous identification of who the leaders in a field are come from? Conferences are only a few times a year; I can't imagine that's where all of these people meet up and socialize. And secondly, how is this mathematician "virtually unknown", as the article puts it? The University of New Hampshire is surely a well-known institution. I'm sure it ranks top 100-500 in the world, right? And there's maybe 20-25 mathematics professors at each university. I also know math is very diverse, and an expert in topology isn't going to know too much about number theory. So within each "domain" of math, there can surely only be a few hundred in the world at any given time who are actively researching the subject. How can he be so unknown then?
- jckt 13y ago@ your second question, the article notes that "Without communicating with the field’s experts, Zhang started thinking about the problem", so maybe he just worked on it in his free time? Not that it means he hasn't published much, but there aren't many Yitang Zhang papers on arXiv either. As far as I know he has no website, and according to UNH he is a "lecturer" rather than a professor. I'm not sure how the academic ranks work in the USA but I believe that, typically, lecturers are of lower "rank" than professors. So in that case, Zhang would have been effectively less renowned, hence "unknown", but obviously no less able.
- curveship 13y agoA lecturer is the bottom of the heap. It's the lowest title you can have and teach in a university. Tenured professor > tenure track professor > visiting assistant professor > adjunct / lecturer. It means he's paid per course and isn't guaranteed any classes beyond the current semester.
- 6ren 13y ago> Without communicating with the field’s experts, Zhang started thinking about the problem. After three years, however, he had made no progress. “I was so tired,” he said. > To take a break, Zhang visited a friend in Colorado last summer. There, on July 3, during a half-hour lull in his friend’s backyard before leaving for a concert, the solution suddenly came to him. “I immediately realized that it would work,” he said. http://c2.com/cgi/wiki?FeynmanAlgorithm http://c2.com/cgi/wiki?FeynmanAlgorithm Write down the problem. Think real hard. Write down the solution.
- mullr 13y agoThis sounds glib on the surface but it's really surprisingly deep. The key is to realize that the problem solving doesn't happen in the second step, but rather in the first. Writing down your problem means you have to structure it in some way, to frame it in a way that at least can be represented on paper. Perhaps it's a formalization, or simply a brief explanation. In any case, step 2 is only possible because of step 1 for most interesting problems. Even for those of us who aren't Feynman.
- ma2rten 13y agoBeen there, done that. It is not as glorious as it sounds. I had some very strong intuitions about some obscure computer science problem (maybe 10 people worked on it worldwide, but I think they gave up already). I was thinking about this problem for at least one year, with some periods of very intensive thinking. I could have really used that time more productively, because that was already quite a busy period for me, but the solution seemed so close. After about one year, I finally had a solution mapped out, which I think would work. However, there were some major implementation problems (memory complexity). That was the time when I was finally able to let it go. So the advise that I would give to people is: focus on real-world problem, that real people have.
- witek 13y agoI disagree. We often don't know what the consequences could be of solving abstract problems by enthusiasts. Whole avenues of science were opened this way.
- banachtarski 13y agoThis is just utterly inspiring and something I've always dreamed about happening to me. Too often, people are proud of themselves after figuring out something that took a few days or even a few weeks to come up with. To stumble upon a solution when you had all but given up hope after three YEARS is just astounding and must amazing. Congratulations to this mathematician. Inspiring. Truly inspiring.
- ky3 13y agothree YEARS Persistence, man, persistence. The chinese have some saying that, loosely translated, "ten years is short to revenge a father's death."
- est 13y ago君子报仇十年不晚 Also this: 10 minutes on stage is 10 years practice behind stage.
- ky3 13y agoNeato, thanks! http://en.wiktionary.org/wiki/%E5%90%9B%E5%AD%90%E6%8A%A5%E4%BB%87%EF%BC%8C%E5%8D%81%E5%B9%B4%E4%B8%8D%E6%99%9A http://en.wiktionary.org/wiki/%E5%90%9B%E5%AD%90%E6%8A%A5%E4... "A nobleman bides his time for the perfect moment for revenge." I swear I didn't make up the oedipalia! A chinese native mentioned it in passing years ago and the phrase stuck.
- nirvanatikku 13y agoThis is simply amazing. Stories like this are humbling, inspiring and provide a renewed appreciation for persistence. Also, with respect to being unknown, the one thing that does pop up upon doing a search is his rating on ratemyprofessor: http://www.ratemyprofessors.com/ShowRatings.jsp?tid=56169 http://www.ratemyprofessors.com/ShowRatings.jsp?tid=56169.
- aarondf 13y agoThe common refrains seem to be 1) easy tests 2) seems nice in class and 3) is mean one in one. I wonder if 2 & 3 have anything to do with his shyness (as mentioned in the article) and his lack thereof when he gets to talking about math? Maybe he gets on a roll as far as math goes, but one-on-one interactions are more difficult? Or maybe he was busy coming up with this bad-ass paper and didn't want to extend his office hours? Either/or.
- pjdorrell 13y agoIs the bound achieved by his proof actually 70000000? That looks like a suspiciously round number. Why can't they tell us the exact bound that he achieved? (Presumably some slightly smaller number?)
- vlasev 13y agoIn math you often don't need super precise bounds. I think it just happens that 70m works. If you try harder you can probably find something more accurate. IMO even a bound like 100m is good because it's a finite number.
- pjdorrell 13y agoNo, actually any proof that supplies a bound must also have some least bound that can be derived from that proof. It could be a calculation that starts from the bound, or it could be some calculation with a result that tells you what the bound is. To me it seems very unlikely that the actual bound was exactly 70000000 in either case. It _could_ be that some super-expensive calculation is required for each candidate bound, and Zhang guessed 70000000 as a starting point and did the calculation for that number, which succeeded, but he didn't bother repeating it for any smaller numbers because it would cost him too much. But if that was the case, that would be interesting in itself.
- ColinWright 13y agoAlso reported six days ago, although with comparatively little discussion: https://news.ycombinator.com/item?id=5703219 https://news.ycombinator.com/item?id=5703219
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- RockofStrength 13y agoThe Twin Prime conjecture is probably the most elementary unsolved mystery in math.
- seanlinmt 13y agoWondering if this impacts the security of public key encryption.
- JackGibbs 13y agoAs part of properly generating RSA keys there are restrictions as to how close together the two primes numbers chosen can be. So no, twin primes bear no relevance to the problem. http://en.wikipedia.org/wiki/RSA_(algorithm)#Faulty_key_generation http://en.wikipedia.org/wiki/RSA_(algorithm)#Faulty_key_gene...