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I almost overlooked this article because I got turned off by the opening description in base 10, as there is a lot of math trivia out there that is specific to

• mixedmath 11y ago
  Indeed, they conjecture that the primes' last digits "conspire" in every base > 2, which is very interesting.
  • knughit 11y ago
    They conspire in base 2 as well, they conspire to be equal. :-)
  • gunnihinn 11y ago
    I'm a mathematician and I'd say their ideas only become interesting if that conjecture is true. Otherwise it's just numerology.
    • ajross 11y ago
      Isn't it sort of a definitional thing about conjectures that they stop being interesting when proven false?
      • indiv0 11y ago
        I think the parent is saying that the statement: Looking at prime numbers written in base 3 — in which roughly half the primes end in 1 and half end in 2 — he found that among primes smaller than 1,000, a prime ending in 1 is more than twice as likely to be followed by a prime ending in 2 than by another prime ending in 1 is not interesting (as it seems to be just numerology), UNLESS the authors' conjecture is also true (that the statement also holds for bases > 2).
        • kafkaesq 11y ago
          The important part is that the authors (and several others) have verified the statistics out to a few hundred billion primes -- and that while the bias does start to drop out, it does so "very slowly." That's what makes this result not "numerology."
        • baddox 11y ago
          It's interesting to me.
        • monochromatic 11y ago
          No, it would be interesting even if it only happened in one base. It's still unexpected.
      • kafkaesq 11y ago
        They might come down a notch, but "failed" conjectures can still be quite interesting, in many ways: https://en.wikipedia.org/wiki/Mertens_conjecture https://en.wikipedia.org/wiki/Mertens_conjecture
      • hiddencost 11y ago
        "not proven true" is not the same as "proven false" they made an observation and a conjecture. If the conjecture is proven false, it's obvious uninteresting. If the conjecture isn't proven either way, it could be argued that it's just apophenia. I'm not sure I agree, but it's not an unreasonable stance.
      • adrianN 11y ago
        P!=NP would still be interesting even if it were proven false.
    • squidfood 11y ago
      Don't they call that the "strong law of small primes": you can discover lots of patterns in the first millions (or now billions) of primes that don't mean anything and don't hold up?
      • mixedmath 11y ago
        I think you're referring to the "strong law of small numbers", as in this excellent article by Guy [1]. But the idea is the same --- so very frequently, patterns that hold for even the first several numbers eventually fail to continue. [1]: https://www.maa.org/sites/default/files/pdf/upload_library/22/Ford/Guy697-712.pdf https://www.maa.org/sites/default/files/pdf/upload_library/2...
    • kafkaesq 11y ago
      I'm a mathematician and I'd say their ideas only become interesting if that conjecture is true. Otherwise it's just numerology. To me, this makes for a very boring notion of "interesting." I think most mathematicians would say that the "interestingness" of a conjecture comes from it (1) describing a phenomenon which seems "intuitively true, or very likely true" (e.g. "x^n+y^n=z^n has no solutions for n>2") combined with (2) the initial difficulty of deciding its truth/falsity using tools available at the time of its statement; along with, finally: (3) the novel techniques (sometime first arising in our brains decades or generations later!) required to ultimately determine said truth/falsity (and the degree to which these techniques touch on and illuminate other areas of mathematics). For example, I think you'd find near-universal agreement among mathematicians that not only would the resolution of FLT (as a conjecture stated my Fermat) would have been equally "interesting" if it had been proven false -- it may have even been more surprising if a counter-example had been found (or its existence proven), provided the tools / lessons were as interesting as those in the Taylor-Wiles result we know today. Meanwhile, some the most interesting conjectures are perhaps those that can't be decided, one way or another. EDIT: If you don't like the idea of discussion the "what-ifs" of a conjecture that's already been decided (like FLT), just plug in any of the usual suspects, e.g. RH or GRH into what I'm saying above. Clearly, a "false" determination on any of these of these major targets -- or even a serious hint at it -- would be career-making achievement for an aspiring mathematician.
      • bradleyjg 11y ago
        >> it may have even been more surprising if a counter-example had been found (or its existence proven), provided the tools / lessons were as interesting as those in the Taylor-Wiles result we know today. What if a counterexample with very large (x,y,z,n) had been found somewhere in the late 1980s because enough megaflops to find it was finally allocated the problem? Would that necessarily have been an interesting result?
        • kafkaesq 11y ago
          Not sure how to answer you on this (because there's a slight chance you might be trolling). Let's just say superficially "yes", in that it would mean the current expert consensus in our universe (that FLT has been proven) would have to be wrong. But it's kind of a bad line of speculation (and so FLT probably wasn't the best illustrative example to bring up in my original post); again, for a real-life instance of a counter-example being found to a conjecture that had a lot of numerical evidence suggestion there wouldn't be one, have a look a the history of the Mertens Conjecture, and others of its ilk. Basic point being that yes, counter-examples to interesting conjectures are always interesting results (and by themselves don't make the original conjecture any less interesting).
    • TTPrograms 11y ago
      It would still be interesting if it holds for all primes rather than just the first ones, as it imposes fundamental structure on the distribution of primes. The base 3 calculation is simply the form that structure takes.
    • jjtheblunt 11y ago
      totally agree.
    • mchahn 11y ago
      > their ideas only become interesting Odds are good that everyone knows the proof that all integers are "interesting". If not, the first non-interesting number would be interesting for being the first. (grin)
    • monochromatic 11y ago
      I think it would still be interesting even if it proved false, simply because it appears to be true for small numbers. That would be weird, and weird is interesting.
    • delhanty 11y ago
      Terry Tao's a mathematician and he seems to think their ideas are interesting: https://terrytao.wordpress.com/2016/03/14/biases-between-consecutive-primes/ https://terrytao.wordpress.com/2016/03/14/biases-between-con...
• vessenes 11y ago
  The article is too vague to assess how interesting the claims are, sadly. It's too bad; I think it wouldn't have detracted from the article to put some more math in. It's not on the face of it at all surprising that sequential primes are more likely to be close to each other modulus any number (3, or 10, or what have you), than they are to be far apart. By way of analogy, a train comes at 1:09pm. Trains come about every 5 minutes between 1 and 2 pm, and only on odd numbers. If you simulate a bunch of random 'next trains', 1 is much more likely than 9 because P(9) approx = !P(1,3,5,7). This is true for all bases. I think what you'd need to be able to say to say something interesting is 1) calculate odds of finding the next prime. 2) Randomly generate numbers with a similar distribution to that of prime occurrence in that range using the Prime Number Theorem at the very least (1 / log(n) probability roughly). 3) check final digits and compare to actual distribution of final digits. If those numbers are very different, then you have in fact found some underlying structure. But the article doesn't hit very hard on this angle, and its hard for (probably) any of us to say just thinking about it with minimal data whether or not there's structure.
  • deleted 11y ago
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  • haberman 11y ago
    This is addressed in the article: > Lemke Oliver and Soundararajan’s first guess for why this bias occurs was a simple one: Maybe a prime ending in 3, say, is more likely to be followed by a prime ending in 7, 9 or 1 merely because it encounters numbers with those endings before it reaches another number ending in 3. For example, 43 is followed by 47, 49 and 51 before it hits 53, and one of those numbers, 47, is prime. > But the pair of mathematicians soon realized that this potential explanation couldn’t account for the magnitude of the biases they found. Nor could it explain why, as the pair found, primes ending in 3 seem to like being followed by primes ending in 9 more than 1 or 7. To explain these and other preferences, Lemke Oliver and Soundararajan had to delve into the deepest model mathematicians have for random behavior in the primes.
    • vessenes 11y ago
• knughit 11y ago
  Primes have no inherent relation to decimal, so all prime patterns in decimal are mathematically meaningful. This is fundamentally different from observations about, say, how 2 and 5 relate to decimal expansions. P.S. never say "base 10", it's a tautology. Say "decimal"
  • macawfish 11y ago
    if "base 10" is a tautology, wouldn't that make "deci-mal" a tautology too? so yeah, just hold up 10 fingers and don't say anything.
    • nsajko 11y ago
      But numerals "10" in bases other than decimal aren't usually called ten, I think.
      • smellf 11y ago
        "10" is never correctly called "ten" except in decimal. But this whole tangent is getting a little navel-gazey for me. "Base 10" is fine in my book because decimal is our inbuilt base.
        • nsajko 11y ago
          That's what I said (or tried to :) ).
      • rlucas 11y ago
        Should we call it "base a"?
        • jholman 11y ago
          "base ||||| |||||" perhaps "base |||||*||", for short.
          • 11y ago
• notthemessiah 11y ago
  Numbers modulo n (i.e. the last digit in base n) are everywhere in number theory and their properties even form the basis of cryptography. What you see as "math trivia" most mathematicians see as mathematics.
  • aidos 11y ago
    I'd guess the GP was pointing out that there are uninteresting results that those less qualified in maths see as some sort of magic. Maybe like those silly "choose any number multiply by x, etc etc did you get 7, magic!" games. Or: 9*9+7 = 88 98*9+6 = 888 987*9+5 = 8888 While not exactly trivial, and there's still maths required to understand the patterns, this prime result is much deeper and more interesting. The GP almost passed over it because they though it might fall into the less interesting category.
    • kafkaesq 11y ago
      Right, but the result in original article is clearly not in this category.
  • BookmarkSaver 11y ago
    That's true, but noticing a phenomenon using an arbitrary base isn't necessarily meaningful. Such as how in base 10 the digits of numbers divisible by 3 or 9 sum up to a number divisible by 3 and 9. There is an underlying principle here, but the specific presentation is just an artifact of the base. This article does a poor job explaining what (or whether) there is some new principle at work here or if this pattern is just a specific presentation of using base 10.
• Houshalter 11y ago
  It doesn't matter if it's specific to base 10 or not. It's still a totally unexpected property. And most math trivia in base 10 does generalize. The numbers change, but the patterns remain the same. But base 10 is far more intuive for people.
  • rickduggan 11y ago
    It's not an unexpected property to me. In particular, I would never have assumed an even distribution of last digit of primes. I'm not sure why this is so magical. It's only so if you assumed it should be evenly distributed to begin with, and there's no particular reason I can think of why this would be so.
    • waterhouse 11y ago
      Then you might be surprised that it's a mathematical fact that the last digit of primes is evenly distributed (among digits coprime to the base), no matter what base you choose. This is one way of stating the Chebotarev density theorem: https://en.wikipedia.org/wiki/Chebotarev%27s_density_theorem https://en.wikipedia.org/wiki/Chebotarev%27s_density_theorem (edit: rephrasing)
      • deleted 11y ago
        [deleted]
      • m00n 11y ago
        Cebotarev is a bit of overkill (and not equivalent!) for this and hard to grasp, if your Algebraic Number Theory is a bit rusty. I recommend Dirichlet's Theorem on arithmetic progressions instead (everybody loves Euler's totient function!) : https://en.wikipedia.org/wiki/Dirichlet's_theorem_on_arithmetic_progressions https://en.wikipedia.org/wiki/Dirichlet's_theorem_on_arithme...
      • rickduggan 11y ago
        Believing that the last digit of primes is evenly distributed overall is not the same as thinking they do not exhibit patterns when viewed sequentially. I'm surprised by neither.
• cecilpl 11y ago
  I wrote a program to count the pairs of adjacent last-digit occurrences in all bases up to 30, for the first 100 million primes, and found this property nearly always holds. Quite interestingly, in all of the few cases where this doesn't hold, primes ending in the digit D are least-frequently succeeded by a prime ending in the digit D-2.
• coldtea 11y ago
  >I almost overlooked this article because I got turned off by the opening description in base 10, as there is a lot of math trivia out there that is specific to base 10 which holds little general significance. If a high school science insight seems to be able to "shoot down" a new scientific discovery, chances are what the discovery says has not really been understood properly. The property they discovered is orthogonal to numeric base.
  • tikhonj 11y ago
    But if high school science insight seems to shoot down a popular math article, it's probably right. Popular articles on technical subjects (all subjects?) simply aren't very good and trivia involving digits in base-10 is exactly the sort of thing that the popular press loves to fawn over. I think that's what the original commenter was getting at: it's not that the actual discovery is suspect, it's that the article presenting it was—even though the discovery itself is interesting.
• jlebar 11y ago
  My experience is that Quanta Magazine specifically is quite good about this stuff in general -- many steps above your regular news outlet.
• omaranto 11y ago
  I almost overlooked this article because I got turned off by the opening description in base 10 The article talks about the last digit: That's also the remainder upon dividing by 10, a perfectly sensible thing to discuss. Similarly what it discusses in base 3 is the last digit, the remainder upon dividing by 3. Properties of numbers, particularly primes, modulo other numbers have been much studied by number theorists.
• dbbolton 11y ago
  Another thing that kind of irked me was this section: >This conspiracy among prime numbers seems, at first glance, to violate a longstanding assumption in number theory: that prime numbers behave much like random numbers... I was under the impression that Ulam's Spiral was a much earlier indication (1960s) that primes weren't really as random as we thought. https://en.wikipedia.org/wiki/Ulam_spiral https://en.wikipedia.org/wiki/Ulam_spiral
• AnimalMuppet 11y ago
  I tested this in base three. For primes below 1000, primes were followed by primes ending in another digit 2.1 times as often as they were followed by primes ending in the same digit. But the ratio shrank as the number of primes grew. By the time I was up to primes below 1,000,000,000, the ratio was only 1.2. That hints that it's an effect that asymtotically disappears...