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.
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.
https://en.wikipedia.org/wiki/Chebotarev%27s_density_theorem
(edit: rephrasing)
I recommend Dirichlet's Theorem on arithmetic progressions instead (everybody loves Euler's totient function!) :
https://en.wikipedia.org/wiki/Dirichlet's_theorem_on_arithme...
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.
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?
I'm curious why floating point operations would be the appropriate tool for finding solutions to a Diophantine equation. Did you have something in mind when writing this? Where can I learn more about it?
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).
https://terrytao.wordpress.com/2016/03/14/biases-between-con...
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)
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
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.
[1]: https://www.maa.org/sites/default/files/pdf/upload_library/2...
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.
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.
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.
> 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.
As I'm writing this out, I'm a little less sure that this would matter, but I'll leave the comment out for the sake of discussion. :)
The first paragraph of the article links to the paper, so people who want more detail can get it. http://arxiv.org/pdf/1603.03720v1.pdf
For your second point, I don't think there's anything wrong with a paper announcing they found something interesting, even if they haven't completely analyzed every aspect of it. Getting the info out early lets a wider audience look at it, and opens their current research up to scrutiny.
By my reading, the article seems to state the key import of the finding quite clearly:
"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."
However this statement:
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'm afraid I don't follow at all. (Do you really mean we should expect that P(1|1) > P(1|9), for either random trains or for subsequent primes? Say wha?)
That said, perhaps you might want to skip straight to the arxiv article itself, or perhaps do some experiments on your own. It's definitely not hard to generate a non-"minimal" amount of data (out to the first few million primes or so) on one's laptop, these days.
9*9+7 = 88
98*9+6 = 888
987*9+5 = 8888
>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.
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.
That hints that it's an effect that asymtotically disappears...
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"
No. There's all sorts of random nonsense in all base expansions of anything that we assign meaning to. Most base-related stuff is numerology.
> P.S. never say "base 10", it's a tautology. Say "decimal"
No, it's not. You need to specify your base number.
so yeah, just hold up 10 fingers and don't say anything.
It's a non-problem though. I don't know about mathematicians, but when programmers say base N they mean N in decimal.
EDIT: woops, replied to the wrong post.
perhaps "base |||||*||", for short.
Peano would be proud.
If you actually believe this, you must be endlessly confused when people say "base 8".
But a little further down, the article discusses how this was discovered originally in base 3, and I think it's much simpler to understand in that context, since all primes except 3 (aka 10 base 3) end in just either 1 or 2:
"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."