
Mills' constant - lelf
http://en.wikipedia.org/wiki/Mills%27_constant
======
bvaldivielso
I find it awesome that they don't even know whether Mills' constant is
rational or not. It certainly looks irrational, but we need a proof to assert
that.

This same thing happens with other constants such as pi + e, pi - e, pi*e, the
Euler-Mascheroni constant... It's good to see that there is yet many things to
learn in order to answer such "elementary" questions

~~~
logicallee
rational? I can't even believe it's for real!

how can a number 1.3063... ^ (3 ^ _any n_ ) produce a prime? (discarding
decimal portion < 1). For _any_ n?

it just seems so super-simple. well, I can think of a simpler constant. is
there one such that raising it to _2_ ^any n produces a prime? (plus decimal
change)?

~~~
alextgordon
There are a countable number of primes, but an uncountable number of real
numbers, and an even more uncountable number of functions over real numbers.

So heuristically, the existence of Mills' constant is really just saying that
primes grow at a relatively consistent rate, which is something that the prime
number theorem formalises.

You can think of Mills' constant as being akin to this real number[1]:

    
    
      0.235711131719232931374143475359616771737983...
    

You could write a program to extract primes from that number, but the
existence of the constant, or the program, is not surprising in the least.

[1]:
[https://en.wikipedia.org/wiki/Copeland%E2%80%93Erd%C5%91s_co...](https://en.wikipedia.org/wiki/Copeland%E2%80%93Erd%C5%91s_constant)

~~~
logicallee
Thanks, this is interesting. As far as writing a program to extract primes
from that number, how could you without including a prime test in the program?
You'd think Erdos and friends would have made it easier by bothering to encode
word boundaries - say with a prefix free code[1] :) - 01 for a word boundary,
00 for a true 0:

0.0120130150170111011301170119012301290131013701410143... (no encoded zeros in
this example.)

Then you wouldn't have to brute-force word boundaries... You could also put
the ordinality of the prime number between 08...09 tags, then you could know
exactly whether you're looking at the millionth prime or the million and
second one.

But yeah other than not encoding boundaries you are right, it's quite akin to
the real number you provided. I like your explanation - thanks.

EDIT: also, as I show above, the program to get a prime back out can seem
complicated. What's remarkable about mill's constant is that the program is
simple: just raise Mill's constant to 3^n (and apply the floor function).
seems so simple. What's remarkable is not just that such a constant _exists_
\- that will give you a prime for _any_ n - but that there is a smallest such
constant and we already know it:

1.3063778838630806904686144926...

I find that remarkable. raise that to any 3^n and you get a prime and change.
wow.

EDIT2: Your argument also didn't address one thing I'm curious about - why a
similar number can't exist for 2^n instead of 3^n... thanks for any thoughts.

[1]
[http://en.wikipedia.org/wiki/Prefix_code](http://en.wikipedia.org/wiki/Prefix_code)

------
taralx
It's not clear to me why the Riemann hypothesis has to hold for the value to
be known.

~~~
gizmo686
From looking at this paper [1], it looks lie the Riemann hypothesis is not
technically necessary, but simplifies an otherwise prohibitively difficult
computation.

[1]
[https://cs.uwaterloo.ca/journals/JIS/VOL8/Caldwell/caldwell7...](https://cs.uwaterloo.ca/journals/JIS/VOL8/Caldwell/caldwell78.pdf)

~~~
scythe
It's because the formula for the maximal prime gap that was used in Lemma 4 at
the bottom of page 3, sqrt(x) log(x) / 8pi, is a consequence of the Riemann
hypothesis. Otherwise the term becomes x^(3/4 + epsilon) which is a lot larger
and messes up the argument. I think it's nontrivially useful here, but there
might still be a proof without it

~~~
gizmo686
From briefly skimming the paper, it looks like the only use of lemma 4 is in
lemma 5, where the author explicitly states that it is possible to prove
without the Riemann hypothesis, if one is willing to work with a bound of
about 10^6000000000000000000.

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curiouslurker
"So what is the application of this?"

I remember a grad student friend in my college days telling me that when
mathematicians present their work, most of their peers never understand it and
this is always the safe question to ask!

~~~
gizmo686
If known, this would allow the easy generation of arbitrarily large prime
numbers, which would win you some cash prizes.

~~~
ikeboy
Prime numbers aren't hard to generate; just generate a bunch of random numbers
and test them quickly.

~~~
Someone
For sufficiently large values of 'large', large primes are hard to generate.
If (that's a big if) we had an efficient method to compute this number to
arbitrary precision, we could compute primes of a billion or more bits.

~~~
czinck
Is there any point to generating primes that large? I thought one of the
reasons to prefer ECC was because it uses a much smaller key than RSA.

~~~
Someone
A) A publication.

B) Eternal fame.

C) Money. See
[https://www.eff.org/awards/coop](https://www.eff.org/awards/coop)

And, most importantly, it may advance mathematics.

