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Comment by dantillberg | Hacker News Reader

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.

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."

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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.

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.

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.

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

(edit: rephrasing)

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_arithme...

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.

Indeed, they conjecture that the primes' last digits "conspire" in every base > 2, which is very interesting.

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.

Isn't it sort of a definitional thing about conjectures that they stop being interesting when proven false?

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).

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."

It's interesting to me.

No, it would be interesting even if it only happened in one base. It's still unexpected.

They might come down a notch, but "failed" conjectures can still be quite interesting, in many ways:

https://en.wikipedia.org/wiki/Mertens_conjecture

"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.

P!=NP would still be interesting even if it were proven false.

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?

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/2...

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.

>> 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?

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).

Thanks for the reply. I assure you I wasn't trolling, I don't know enough about the topic to construct a troll even if I wanted to.

No worries -- your question was totally reasonable (and an in fact, a perfectly good question).

> 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

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?

I think he just means processing power, generally.

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.

totally agree.

> 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)

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.

Terry Tao's a mathematician and he seems to think their ideas are interesting:

https://terrytao.wordpress.com/2016/03/14/biases-between-con...

>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.

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.

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.

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.

They did mention this, but they didn't talk real numbers. And, my second point is (I think) slightly more subtle -- the probability distributions need to be considered, not just the counting upward angle.

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. :)

I don't know why you're nitpicking. The article's written for a more general audience that may be interested in the property, but not necessarily the nitty-gritty math behind it.

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.

The article is too vague to assess how interesting the claims are, sadly.

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.

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.

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.

Right, but the result in original article is clearly not in this category.

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.

My experience is that Quanta Magazine specifically is quite good about this stuff in general -- many steps above your regular news outlet.

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

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.

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...

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"

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.

But numerals "10" in bases other than decimal aren't usually called ten, I think.

"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.

That's what I said (or tried to :) ).

Should we call it "base a"?

"base ||||| |||||"

perhaps "base |||||*||", for short.

base II*II+II?

base X (romans)

Ten is "decem" in Latin. Decimal means, literally, base ten. It's a tautology either way. You can't even specify the base of the base since "base (10 in base 10)" presents the same problem recursively.

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.

I get the parent poster's point, but to me it's like quibbling over people using "comprised of" instead of "comprises": some people will care, but in the grand scheme of things it's not a meaningful disagreement because everybody got what you were trying to say the first time, and there are only a few rare contexts where the distinction matters.

Base (((((((((1++)++)++)++)++)++)++)++)++)

Peano would be proud.

> Primes have no inherent relation to decimal, so all prime patterns in decimal are mathematically meaningful.

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.

I'm not certain, but I believe the subtle point parent was trying to make, is the difference between "base 10" and "base ten". Every base is "base 10" in its own base ;)

> P.S. never say "base 10", it's a tautology.

If you actually believe this, you must be endlessly confused when people say "base 8".

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