
Mathematicians Discover Prime Conspiracy (2016) - yurisagalov
https://www.quantamagazine.org/mathematicians-discover-prime-conspiracy-20160313/
======
krick
> 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.

Ok, I verified on 10 000 tosses, it works, but I cannot see why. Can somebody
explain?

Edit: nevermind, it can be found in the previous discussion.

~~~
vegizombie
For anyone else interested, the way this was explained to me a while ago was
to look at the failure mode after an H. For Alice, her failure for HT is HH,
therefore her next flip can land on T, completing the sequence. For Bob, his
failure mode is HT, so he now needs to flip a H before he can try for the 2nd
H.

~~~
boi4
Related:
[https://en.wikipedia.org/wiki/Knuth–Morris–Pratt_algorithm](https://en.wikipedia.org/wiki/Knuth–Morris–Pratt_algorithm)

~~~
natmaka
Related and well-known by software developers:
[https://en.wikipedia.org/wiki/Boyer%E2%80%93Moore_string-
sea...](https://en.wikipedia.org/wiki/Boyer%E2%80%93Moore_string-
search_algorithm)

------
air7
I find number theory to be closer to a natural, investigative science akin to
Physics rather than pure math such as calculus.

IANAM, but I doubt there was ever a modern calculus theorem that was hinted by
a brute force search. One can argue that Calculus is "made up" and thus
research in this field is finding deeper logical consequences for the initial
axioms, while natural numbers just "have properties" and research is Number
Theory is trying to uncover these properties by brute force and later trying
to prove them, arguably quite unsuccessfully so far. It's an interesting
tidbit in the philosophical question of "Does math exist only in our head?"

~~~
qubex
Your intuition is not all that wrong, but it’s not entirely correct either.

Number Theory is basically the exploration of the deep consequences of a set
of (ultimately arbitrary but apparently obvious) axioms. The exploration of
those consequences arises from the fact that they are ultimately too
plentiful, ramified, and incompletely provable (per Godel’s Incompleteness
Theorems & Turing’s Halting Problem).

~~~
rumanator
> Number Theory is basically the exploration of the deep consequences of a set
> of (ultimately arbitrary but apparently obvious) axioms.

Isn't that true for all fields of pure math?

~~~
chess93
Almost all fields of pure math use a much stronger set of axioms called ZF and
essentially everyone also accepts the axiom of choice (making it ZFC). The
axioms in ZF are reasonable but the axiom of choice is surprisingly
controversial for an axiom. There are some unintuitive consequences of the
axiom but even more unintuitive consequences without it or with the negation
of it.

Number theory uses a much smaller set of axioms.

It should be stated that most mathematicians don't really mind the logical
foundations of their work when they are actually working in the same way that
most programmers don't worry about assembly language or transistors.

------
ganzuul
Abstract: While the sequence of primes is very well distributed in the reduced
residue classes (mod q), the distribution of pairs of consecutive primes among
the permissible ϕ(q)2 pairs of reduced residue classes (mod q) is surprisingly
erratic. This paper proposes a conjectural explanation for this phenomenon,
based on the Hardy-Littlewood conjectures. The conjectures are then compared
to numerical data, and the observed fit is very good.

\- [https://arxiv.org/abs/1603.03720](https://arxiv.org/abs/1603.03720)

------
dang
Discussed at the time:
[https://news.ycombinator.com/item?id=11282480](https://news.ycombinator.com/item?id=11282480)

~~~
Terr_
LOL, after posting my other reply I visited your link... and I just realized I
repeated myself from 3 years ago.

I'm not sure how to feel about that reminder of age.

~~~
kristiandupont
Two days ago I found an elaborate response to something I was researching on
Reddit, only to discover at the bottom that I was the person who wrote it, 5
years ago. It's a weird feeling indeed!

~~~
Already__Taken
I've been getting stack overflow answers every blue moon to my node proxy
settings question for nearly 10 years. Weird feeling, it's nice to know we
exist after we look away.

------
lancebeet
>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. Most mathematicians would have assumed, Granville and Ono
agreed, that a prime should have an equal chance of being followed by a prime
ending in 1, 3, 7 or 9 (the four possible endings for all prime numbers except
2 and 5).

This seems like an odd assumption to me. Surely sexy primes are more common
than twin primes, so at least for primes that are near each other there should
be a higher probability for certain sequences of final digits. This is
obviously not proof in itself, but it would certainly make me hesitate to
assume there is an equal probability in the general case.

~~~
soVeryTired
> Surely sexy primes are more common than twin primes, so at least for primes
> that are near each other there should be a higher probability for certain
> sequences of final digits.

I think that's conjectural, but prime constellations are also conjectured to
be a negligible fraction of primes as a whole, much as primes are a negligible
fraction of integers (asymptotically). I think twin primes are conjectured to
be distributed as n / (log n) ^2, while primes are n / (log n).

Besides, even if (p, p + 6) is significantly more common than (p, p + 2), if
the final digit of p is uniformly distributed among 1, 3, 7, and 9, I don't
think the statistics as a whole are affected.

------
blackrock
I always wondered if prime numbers can be used as the basis of a compression
algorithm.

Where you describe how to get your number, in relation to how it can be
factored using prime numbers.

You cut up your data into chunks, and then find the prime factors of each
chunk. Now you have a list of arithmetic descriptions to describe your data,
and you can use Huffman to compress the descriptions.

Of course, this would be computationally intensive, for both the compressor,
and the extractor, but it may save on the payload size, to allow for efficient
data transfer.

And while we still use Huffman’s algorithm, I wonder if advanced aliens may
have discovered this technique for data compression.

This is more of a sci-fi fantasy algorithm, than anything actually
mathematically sound.

~~~
Simon_says
The prime factors of a number takes (about) the same amount of space to list
as the number itself.

~~~
saagarjha
Notably, even listing the ordinal index of the prime (e.g. that a number is
the product of the first, fifth, and tenth primes) takes about the same amount
of space as well due to the prime number theorem.

~~~
yholio
> the average gap between consecutive prime numbers among the first N integers
> is roughly log(N)

So if you are compressing exceptionally large numbers, say, a whole video
file, which is likely to contain very long factors, you could shave
log2(log(N)) bits for each factor of length N. Seems like the most impractical
compression algorithm imaginable.

------
bmn__
paper: "Unexpected biases in the distribution of consecutive primes"
[http://arxiv.org/abs/1603.03720](http://arxiv.org/abs/1603.03720)

explanation for non-experts: "The Last Digit of Prime Numbers - Numberphile"
[https://www.youtube.com/watch?v=YVvfY_lFUZ8](https://www.youtube.com/watch?v=YVvfY_lFUZ8)

~~~
arberavdullahu
Is the Numberphile video based on the paper? Because the paper was out on 30
May 2016 and video was out on 4 May 2016

~~~
bmn__
Read the paragraph with the heading "Submission history".

------
siliconunit
Classic example of following definitions or illustrious assumptions too
closely... another one is an obvious one, and historically has been challenged
a few times: the beginning of the prime sequence 'should be' 1, 3, 5.. very
clean, very coherent, but it got stuck in definitions limbo and now we got no
1 and the alien 2, that has to be excepted in most algorithms..eh...

~~~
lanna
Why do you think 2 is alien? How could the fundamental theorem of arithmetic
be stated without 2?

~~~
ianai
2 is definitely no “alien”. More like a legendary prime god amongst countably
infinite odd primes. 0 and 1 are two sides of an extreme that either generates
a meaningful number system through their separation and extremis from one
another or no system at all if 1=0 is assumed true. They’re like a Jekyl and
Hyde god.

~~~
lanna
"Odd" is just a word that means "not divisible by two". There is truly nothing
special about a prime not being divisible by two. I could equally say "3 is a
legendary prime god amongst countably infinite non-divisible-by-three primes",
"5 is a legendary prime god amongst countably infinite non-divisible-by-five
primes"... The only difference is that we do not have special words that mean
non-divisible-by-three, non-divisible-by-five, etc.

In this regard, I don't see 2 as a special prime. Saying "2 is the only even
prime" is the same as saying "2 is the only prime divisible by two", which, by
definition, can be said about any other prime.

~~~
ianai
Right. In the language of the article, if p is a whole number whose digits end
in the number 2 then all numbers after it ending in the digits 0 or 2 are not
prime. It’s a seemingly weak statement but it says a much stronger thing about
the primes than the hypothesis of the article.

Anyway, just trying to react in tone to calling any number an “alien”. I think
of numbers as being outside the realm of words like alien - I don’t even know
what that would mean for a number. Hence my comment is hyperbole hinting at my
love of mathematics. To me, it really is a beautiful area of study. Wish I
could do it more.

------
joantune
I wonder what the implications are for cryptography - given the heavy reliance
on 'random' prime numbers

but one would have to delve into this - and from the top of my head there are
no algorithms that make you pick 'two consecutive' primes - so the
implications - if any - aren't obvious - anyone here has some more light or
other angles to shed into this?

~~~
foo101
Why would this have any implications on cryptography? This does not make
factoring an RSA modules any easier.

~~~
joantune
Well, in RSA you have to choose two prime numbers, multiply them, and keep
them secret. p and q: pq =n . And n is made public. I wonder if this
probability, maybe coupled with concrete implementations, makes it for a more
restricted set of guessing p and q. That would be it. I guess that given that
this only introduces 'restrictions' on __sequential __prime numbers, doesn 't
really help at all, given that p and q should be random. Unless there's a
shortcut applied in implementation that you find a random p and then q is the
next prime number. Hence my question to the community. But I only know that
both RSA and DH rely on prime numbers.

~~~
foo101
That's a fair point. Yes, I thought the same thing too when I read this. It is
only introducing restrictions on sequential prime numbers. I doubt it has any
bearing on the factoring problem yet, at least not without more work that can
connect this to the factoring problem.

------
ngneer
Not sure I understand the significance of a last digit of 9 for a prime
implying the next prime is likelier to end with a 1 and not a 9. After all,
base 10 is arbitrary. Can someone explain whether similar such "conspiracies"
occur for all bases?

~~~
Tyr42
Yeah, the article says they investigated in base 3 first.

In that base, it can by 0, 1, or 2 mod 3 (for the final digit). Primes clearly
aren't 0 mod 3, so all (not 3) primes end in 1 or 2. They found that primes
"liked" to alternate between them.

------
Terr_
I wonder if it's a some extension or variation on Benford's Law [0].

[0]
[https://en.wikipedia.org/wiki/Benford%27s_law](https://en.wikipedia.org/wiki/Benford%27s_law)

~~~
acvny
I don't think this is a law and it's stupid. It is just an observation. Of
course 1 will be more frequent.

~~~
peheje
Why of course

~~~
sjnu
Because 100 to 200 is twice as much and 700 to 800 is not

~~~
joantune
Carry on.. why does that make it obvious?

------
entwife
In base 30, a prime greater than 30 can end in only 9 of the possible 30
terminal digits. In contrast with base 10, where this statistic is 4 of 10.

