4 ms·

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...

They conspire in base 2 as well, they conspire to be equal. :-)

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...