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## Mathematicians Discover Prime Conspiracy (quantamagazine.org) 227

An anonymous reader writes with an intriguing story at Quanta Magazine, which begins: Two mathematicians have uncovered a simple, previously unnoticed property of prime numbers — those numbers that are divisible only by 1 and themselves. Prime numbers, it seems, have decided preferences about the final digits of the primes that immediately follow them. Among the first billion prime numbers, for instance, a prime ending in 9 is almost 65 percent more likely to be followed by a prime ending in 1 than another prime ending in 9. In a paper posted online today, Kannan Soundararajan and Robert Lemke Oliver of Stanford University present both numerical and theoretical evidence that prime numbers repel other would-be primes that end in the same digit, and have varied predilections for being followed by primes ending in the other possible final digits. "We've been studying primes for a long time, and no one spotted this before," said Andrew Granville, a number theorist at the University of Montreal and University College London. "It's crazy."
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## Mathematicians Discover Prime Conspiracy

• #### Forget something? (Score:5, Informative)

on Monday March 14, 2016 @12:54AM (#51691923)
• #### Re:Forget something? (Score:5, Insightful)

on Monday March 14, 2016 @12:57AM (#51691935)

The link isn't in the summary -- but off to the right of the title.

I've hated this "feature" of /. every since they implemented a year or so ago.

• #### Re: (Score:2)

It is the lazy designer's way of handling it -- a field on the submission form rather than putting it in as a link under the words "A new study suggests..."

They could create an internal tool to highlight some words in the OP text when approving, hit a button, and boom, the text contains the link.

That would involve them at least nominally being web board developers. Oh, wait.

• #### Re: (Score:2)

More annoying, the "off to the right" link doesn't appear on the mobile site at all. Browsing from mobile there's no way to RTFA unless someone links it in the comments.

• #### Twim primes? (Score:5, Interesting)

on Monday March 14, 2016 @01:12AM (#51691961)

I wonder if this has anything to do with Twin primes [wikipedia.org]. If a prime ends in 9, then its twin will end in 1, and so we should expect primes ending in 9 to more often be followed by primes ending in 1. The number of twin primes is believed to be infinite, but they get more sparse as you go towards infinity (proportional to 1/(ln(n)^2)), even faster than primes (proportional to 1/ln(n)), so if they are responsible for the bias, then the bias should diminish as you go up.

• #### Re:Twim primes? (Score:5, Interesting)

on Monday March 14, 2016 @02:10AM (#51692075)

I wonder if this has anything to do with Twin primes.

Yes, they are most likely related. Both the twin prime conjecture and these results about the final digits can be derived from the prime k-tuple conjecture. Or so says the fine article. It is not immediately obvious to me why the current result is predicted by the prime k-tuple conjecture but it does sound reasonable.

• #### In the digits of pi (Score:2)

Did you know that all prime numbers appear in consecutive digits of pi, along with the complete works of shakespear?

Pick any statistical anomaly and it likely that some of these will appear over some run of an unpredictable series.

• #### Re: (Score:2)

Not necessarily. Even though PI isn't a rational number, it doesn't mean that it necessarily contains any given subsequence. Let me take another non-rational number: 0.12112111211112... It won't ever contain the prime number 3, nor will it contain the complete works of Shakespeare. PI is similarly constrained to a specific pattern. I agree that an infinite un-patterned sequence will contain such sequences, but whether you include or exclude the axiom of choice will determine whether such numbers exist o
• #### Re: (Score:2)

I wonder if this has anything to do with Twin primes [wikipedia.org]. If a prime ends in 9, then its twin will end in 1, and so we should expect primes ending in 9 to more often be followed by primes ending in 1.

If a prime ends in 7, then its twin will end in 9. Without further detail, we'd expect half the 9 primes to be followed by a composite. Your explanation is missing something. Why should 9-1 happen more often than 7-9?

• #### Re: (Score:2)

It does diminish: "The biases that they found appear to even out, little by little, as you go farther along the number line — but they do so at a snail’s pace."

• #### 8 9 (Score:3, Funny)

on Monday March 14, 2016 @12:54AM (#51691925)
7 did it.
• #### Cut it out! (Score:5, Funny)

on Monday March 14, 2016 @12:57AM (#51691929)

Stop anthropomorphizing prime numbers. They hate that!

• #### Re:Cut it out! (Score:5, Informative)

on Monday March 14, 2016 @01:56AM (#51692037)

And stop linking to the news article only, without linking to the scientific paper. Just for those who care, here is the link: http://arxiv.org/pdf/1603.0372... [arxiv.org]

• #### Re:Cut it out! (Score:4, Informative)

<uninvolved@outlook.com> on Monday March 14, 2016 @09:22AM (#51692893) Journal

WTF? Why, in the name of all that's good, would they...

Oh, I just noticed. You're still new here. *sighs*

Look, nobody reads the article. Nobody is going to read a scientific paper. Well, a few of us might read the article (I'm not admitting to anything) but those of us who do, also know how to find the applicable paper.

If you look at the very top post in the thread, there's someone bitching that there is no link to the article. Yet, the article link is right next to the title - where it has been for almost a year now. (They're sometimes in the summary as well. Not always.) That should tell you, they being a representative of the average one of us, how often we actually even read the article - or even look for the URL.

They're not going to do it. The two other people who read the article know where Arxiv is. The editor would have to, you know, work. Ain't happening. Submit stories with the link included if you're passionate. 'Snot going to change in your lifetime. You're probably the 10,985,729th (see what I did there?) person to suggest that - this month.

• #### Re: (Score:2)

Don't say that, you're going to make the prime numbers angry!
• #### What other bases does this hold for? (Score:5, Interesting)

by Anonymous Coward on Monday March 14, 2016 @01:01AM (#51691943)

Only 65%? Pft. In base 2, every prime number is 100% likely to be followed by a prime ending in 1.

• #### Re:What other bases does this hold for? (Score:5, Insightful)

on Monday March 14, 2016 @01:18AM (#51691969)
>> In base 2, every prime number is 100% likely to be followed by a prime ending in 1

That was kind of my thought too. Isn't the "9/1" thing kind of base 10-ist?
• #### Re:What other bases does this hold for? (Score:4, Interesting)

on Monday March 14, 2016 @01:27AM (#51691981)
TFA reads like buzzfeed of number theory, when high schoolers get all excited about pop-sci.

Cyclic groups and observable symmetries in there are well studied field. In this particular case, it's about primes projected on a modulo 10 group. There are thousands of those exhibiting various biases, yet this one is somehow exciting because it coincides with decimal base.
• #### Re:What other bases does this hold for? (Score:5, Informative)

on Monday March 14, 2016 @02:00AM (#51692047)
If you rtfa it says "Lemke Oliver and Soundararajan discovered that this sort of bias in the final digits of consecutive primes holds not just in base 3, but also in base 10 and several other bases; they conjecture that it’s true in every base. "
• #### Re: (Score:2)

Ava K Lamb explains the cultural significance of prime numbers best:

https://youtu.be/_inzEWQRRsY?t... [youtu.be]

• #### Re: (Score:2)

Probably. The conjecture needs to be stated more generally and then tested...

Conjecture: For any prime ending in the largest odd digit for the base system, there is a 65% chance that the next prime will end in the first non-zero whole digit for the base (e.g. the number 1).

In other words....
What is the chance that the following holds true?
Base 2: Last prime ends in 1, next prime ends in 1
Base 3: Last prime ends in 2, next prime ends in 1
Base 4: Last prime ends in 3, next prime ends in 1
Base
• #### Re:What other bases does this hold for? (Score:4, Interesting)

on Monday March 14, 2016 @02:29AM (#51692109)
No joke, I just googled the exact same thing. Apparently prime numbers are prime in all bases, which I really didn't think was the case.
• #### Re:What other bases does this hold for? (Score:4, Informative)

on Monday March 14, 2016 @02:54AM (#51692139) Journal

Why would you think that? The laws of division don't change for different base representations. Division is division no matter how you write the number.

• #### Re: (Score:2)

Why would you think that? The laws of division don't change for different base representations. Division is division no matter how you write the number.

Because an awfully lot of "cool math tricks" have more to do with division mod 9/10/11 than any actual underlying structure of the numbers themselves. If this only worked in base 10, I'd tend to dismiss it as just another "gee how cool you just learned the pigeonhole principle" trick. But since it works in at least a few other bases, it suggests the pre
• #### Re:What other bases does this hold for? (Score:5, Insightful)

on Monday March 14, 2016 @05:00AM (#51692331) Homepage
The base doesn't change what the number IS, only how it is written down.
• #### Re: (Score:2)

Numbers are numbers, they exist and have properties independently of their representation.

• #### Re: (Score:2)

Hadn't thought about it before but it makes sense. Seven is prime whether it's 7 or 0111. Base is just representation after all.
• #### Re: (Score:2)

No joke, I just googled the exact same thing. Apparently prime numbers are prime in all bases, which I really didn't think was the case.

Surely you jest. Numerical bases are just a convenient way to represent numbers. Numbers, and their properties, including primality, exist independently of the base in which we represent them. A number does not stop being prime or odd or even just by changing the way we encode them.

• #### Re: (Score:3)

Ah, grasshopper... When you understand *why* then you'll start to understand maths. It makes sense that it does so, does it not?

I'm an honest-to-goodness real-life one-of-them-there mathematicians (fully papered and everything) and I'll share the silliest thing that I can think of.

I hated math. No, I hated it. It made no sense - but I was good with rote. Then, a teacher shared something along the lines of this, "You know, if you just square the triangle and divide it in half, it's the same thing."

Now that m

• #### Re:What other bases does this hold for? (Score:4, Informative)

on Monday March 14, 2016 @10:02AM (#51693113)
There are very few numerical properties that are base-dependent.

Some of the little tricks they teach you in school are strictly base-dependent, like if a decimal number ends in 5 or 0 it's divisible by 5 or 10 respectively. If a decimal number ends in a value divisible by 2 it's even else odd. Or if a decimal number's digits sum to a multiple of 3 or 9 then it's divisible by 3 or 9 respectively.

What they don't tell you is that is generalizable to other bases. Generically speaking if the final digit of a number in a given base is divisible by any factor of that base then the number itself is divisible by that factor (this should be fairly obvious) and if the digits of a number sum to a number divisible by a factor of (base-1) then that number itself is divisible by that factor (less obvious, but provable).

So for hex, for example, the factors of 16 are 2, 4, 8, 16. If a number in base-16 ends in 0 it's obviously divisible by 16, if it ends in 8 then it's divisible by 8 and so on. The factors of (16-1)=15 are 3, 5, and 15. So if the sum of digits of a hex number are divisible by 3, 5, or 15 then the number is also divisible by 3, 5, or 15 respectively as well.

Fun little math quirks on bases.
• #### LOL (Score:4, Funny)

by Anonymous Coward on Monday March 14, 2016 @01:11AM (#51691959)

>Two mathematicians have uncovered a simple, previously unnoticed property of prime numbers — those numbers that are divisible only by 1 and themselves.

Did anyone else LOL when they read the first sentence. My first thought was who wouldn't notice primes are only divisible by 1 and themselves it's the definition, duh.

• #### Re:LOL (Score:5, Funny)

on Monday March 14, 2016 @01:22AM (#51691973)
>> Did anyone else LOL when they read the first sentence.

Yes. I initially though someone had pranked SlashDot by convincing the editors that no one knew that property of primes before. If so, that would have been the ultimate SlashDot dup - 2500 years or so in the making.
• #### Re: (Score:2, Funny)

by Anonymous Coward

Some readers may not understand the term "prime number", so an explanation could be very useful.

I am currently reading Slashdot on a "computer", which is an electronic device that stores and manipulates information.

• #### Re: (Score:2)

Dude, every kid knows what a computer is. Seriously, where do you think we are?

• #### Re: (Score:2)

You are not alone. My first thought was, "Man, the Slashdot reader has gone downhill." My second was, "Oh, we're being baited. Makes sense from a business perspective. I'd be surprised if BIZX wasn't doing so intentionally. They'd almost be silly to not do so. We're pretty gullible and love outrage."

Then I realized that it was probably a little of both and here we are.

At any rate, it makes good business sense to include trivial things like that. It gets people talking. There's almost always something for th

• #### perhaps true of other sequences? (Score:2)

Take other, random sequences of integers with the same density within the integers. I wouldn't be surprised if you find similar anomalies.
• #### Get off my lawn! (Score:2)

Get off my lawn! says prime number to its siblings.

present both numerical and theoretical evidence that prime numbers repel other would-be primes that end in the same digit, and have varied predilections for being followed by primes ending in the other possible final digits.

• #### Bizarre paragraph in the linked article (Score:3)

on Monday March 14, 2016 @02:27AM (#51692101)

"Soundararajan was drawn to study consecutive primes after hearing a lecture at Stanford by the mathematician Tadashi Tokieda, of the University of Cambridge, in which he mentioned a counterintuitive property of coin-tossing: If Alice tosses a coin until she sees a head followed by a tail, and Bob tosses a coin until he sees two heads in a row, then on average, Alice will require four tosses while Bob will require six tosses (try this at home!), even though head-tail and head-head have an equal chance of appearing after two coin tosses."

This seems wrong to me. Both Alice and Bob have equal chance of rolling a head, hence on average they will need the same number of tries to arrive at a head toss; and since coins dont have memory, the next toss has equal chance of being head or tail. So I do expect the chances of head head and head tail to be the same.
This is how I see it, the statement above came from a mathematician so I am probably making a mistake in my reasoning but I don't see where.
Any input?
• #### Re:Bizarre paragraph in the linked article (Score:5, Interesting)

on Monday March 14, 2016 @02:41AM (#51692129)

This seems wrong to me. Both Alice and Bob have equal chance of rolling a head, hence on average they will need the same number of tries to arrive at a head toss; and since coins dont have memory, the next toss has equal chance of being head or tail. So I do expect the chances of head head and head tail to be the same.

Yes, it's called a veridical paradox. That's something that seems impossible but is nonetheless true. You can verify it by flipping a coin, or running a computer simulation using a good random number generator.

• #### Re: (Score:2)

But we should be able to calculate this instead of trying it. So I understand where my error is.
• #### Re:Bizarre paragraph in the linked article (Score:5, Insightful)

by Anonymous Coward on Monday March 14, 2016 @03:37AM (#51692207)

Intuitively it makes sense. Assume the first H has been tossed. For Alice, she fails by tossing another H. However, this second H can be the first H of a successful HT sequence, so in failure there is a silver lining - she's halfway to success and can stop after tossing a single T. Full sequence: HHT.

For Bob, after tossing the first H, tossing a T means he has to start over. He needs to toss another H first, followed by yet another H to succeed. His task is harder. Full sequence: HTHH.

• #### Re: (Score:2)

yes, I get it completely now. thanks.
• #### Re:Bizarre paragraph in the linked article (Score:4, Funny)

on Monday March 14, 2016 @09:43AM (#51692991)

You missed the absolutely critical corollary that restores balance to the force: after Bob succeeds, he's already halfway to his next success where after Alice succeeds, she needs to snooze for one toss before she's back in the game, where apparently the game involves some gender-swap role play.

It's so totally male to cease thinking the problem through after attaining the initial success condition.

I think I could teach a very interesting grade XI math class.

Corollary: I would end up behind bars.

• #### Re: (Score:2)

The AC intuitive explanation is correct, and you can calculate it by drawing the tree.
After 4 rolls, I get that Alice probability to succeed is 11/16 (1/4 for HT, 1/8 for HHT and THT, 1/16 for HHHT, THHT and TTHT), while Bob probability to succeed is 8/16 (1/4 for HH, 1/8 for THH, 1/16 for HTHH and TTHH)

• #### Re: (Score:2)

Another way to see it is to look at the pattern formed by a completed sequence.

For Alice, it is T* H+ T.
For Bob, it is T* H (T+ H)* H.

• #### Re: (Score:2)

You can verify it by flipping a coin, or running a computer simulation using a good random number generator.

Yup, I got 4.00171 for "HT" and 5.999467 for "HH" after a million iteration.
Fun stuff!

• #### Re: (Score:2)

Fascinating. Does that particular paradox have a name? It reminds me of Monty Hall's game paradox.

• #### That's nothing (Score:4, Funny)

on Monday March 14, 2016 @02:30AM (#51692111) Homepage

After a prime ending in 2 or 5, there has to be at least a billion primes before another one can end in 2 or 5.

Shachar

• #### Re: (Score:2)

going by the definition in TFS, -2 would qualify as a prime. And base 3 numbers would also mess up the theory.
• #### Re: (Score:2)

I'm sure I'm missing a hilarious joke but a number ending in 2 would be even and not prime...
• #### Re: (Score:2)

You are - but it's not hilarious.

2 and 5 are both only primes once. After that, any number ending in either is no longer a prime. Thus, all prime numbers after that (at least a billion) will not end in either a 2 or a 5. It's a knee slapper.

• #### Re: (Score:2)

You're not missing it. 2 and 5 themselves (in decimal, or any base where they are factors of the base) are the only prime numbers ending in 2 and 5. So while GP is correct in a logical sense (there are at least a billion numbers afterwards ending in neither 2 nor 5 that are also prime) there are, in fact, infinite numbers afterwards ending in neither 2 nor 5.

This is not true in base-16 for 5's however (0x25 = 37 is prime for example), but remains true for 2's. It's completely false for base-7 though (ba
• #### Is that a constant? (Score:2)

By the general nature of extremely basic things in mathematics, shouldn't "almost 65%" actually be a number equal to some sort of constant or famous sequence? Or there's a pattern with the 2nd to last digit then 3rd to last digit etc that assembles some sort of other known constant. Since prime numbers are sort of like a constant and there's a lot of constant crossover between equations and each other, you would think something like the Fibonacci sequence or other common constant would show up in a situati
• #### Re: (Score:2)

Maybe you're thinking of fibonacci/golden ratio type thing.
• #### Re: (Score:2)

Rough morning? They mention Fibonacci in their post. ;-) So yes, yes that is what they're probably thinking of. Gonna be a long Monday.

• #### Psshh they ALL end in 1 (Score:4, Interesting)

on Monday March 14, 2016 @02:44AM (#51692131)

What's this base 1010 drama? Everyone knows in binary ALL primes end in "1".

Jokes aside, the fact that there's plenty of bases to choose from means that what they are really talking about is the modulo remainders of primes having a pattern- and modulo division and primes have had a pretty flirty relationship. Unquestionably interesting. The thing with the prime number set is that it's immutable- a set of fixed numeric stars shining the same light since before time began, and yet even with that constancy, many functions involving the prime number web have proven frustrating to calculate for large values- there's hardly any shortcuts compared to the integer math you run into on a daily basis.

• #### Re: (Score:3, Informative)

> What's this base 1010 drama? Everyone knows in binary ALL primes end in "1".

In binary ALMOST all primes end in "1".

• #### Re: (Score:2)

Everyone knows in binary ALL primes end in "1".

10.

• #### The occurrence of digits is linked to the base (Score:3, Funny)

on Monday March 14, 2016 @03:28AM (#51692199)
Last time I looked at primes in binary I noticed a 100% chance that the next one ended in 1. No I am not trolling you I'm just making a point, go look at the primes in different bases and see what you notice.
• #### So what's the name for being base 10 prejudiced/ (Score:2)

And can we establish a university department to run courses challenging it? I get to be the overpaid professor in charge as I thought of the idea, but I'm sure that there will be plenty of teaching posts focusing on the need to avoid such an elementary form of discrimination...
• #### Early april first joke? (Score:2)

Of course a prime ending in 9 will most often be followed by a prime ending in 1.

Primes are more-or-less "random": you can't easily predict the next prime. In every number-range primes have a certain density, and that density drops as numbers get larger. So "around 1000", the prime-density is higher than "around 10000".

So assuming the prime density is P and we have a prime p ending in nine, there is a P chance of p+2 (ending in 1) being prime. Then there (1-P)P chance of the next candidate being prime. Th

• #### I wonder how many already stumbled upon this (Score:2)

I wonder how many already stumbled upon this and just assumed that it must be already mentioned somewhere.

For instance if I noticed that the probability of a prime ending in a 9 went up and down in a sign wave over the first billion primes, I would again just assume that this was a well known fact and move on.

So I wonder how many of these previously ignored discoveries are going to be dusted off now that people have been reminded that there are fundamental discoveries still unclaimed with primes. Also
• #### Related to Benford's law? (Score:2)

I'm wondering if this isn't merely a correlation to Benford's law being applied to the prime set. IANAM (not a mathematician) but this kind of pattern in primes doesn't seem to be particularly counter-intuitive to me on the face of it.

• #### Call it Jenny's Theorem (Score:2)

Just to entice some kids onto my lawn, I'll point out that my favorite twin primes are a good example of this. Once you meet Jenny it's hard to forget her. Linky [wikipedia.org]

• #### hyperbole (Score:2)

"We've been studying primes for a long time, and no one spotted this before," said Andrew Granville, a number theorist at the University of Montreal and University College London. "It's crazy."

I imagine the number theorist in a shower cap screaming "it's crazy as hell!".

http://stream1.gifsoup.com/vie... [gifsoup.com]

• #### It's PI DAY (Score:2)

Happy PI Day /.

#### Related LinksTop of the: day, week, month.

It is clear that the individual who persecutes a man, his brother, because he is not of the same opinion, is a monster. - Voltaire

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