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Mills' constant

en.wikipedia.org

44 pointslelf20 comments

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

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)?

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...

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

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

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...

It also surprised me how simple the more general proof https://primes.utm.edu/notes/proofs/A3n.html is (except for that pesky invocation of the Riemann hypothesis to prove that there is a prime between successive primes)

But (pedantic): it may be that we use a cannon to kill a fly only because we don't know yet that the thing we attempt to kill is a fly, but it appears we don't know that for sure, either. Maybe it is a bullet-proof fly the size of an elephant (would be a cool result: The Riemann hypothesis is true iff, for all N, there is at least one prime between N^3 and (N+1)^3)

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

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.

"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!

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

It might seem like that based on the definition, but read the whole article and you'll find it's the other way around.

Mill's number is shown to exist be showing that primes in certain ranges exist, and then building the number (somewhat like constructing a number from its decimal expansion).

So you don't use Mill's number to construct prime numbers, you use prime numbers to construct Mill's number.

The EFF Cooperative Computing Award rules, by the way, specifically say that you can't just claim (say) A^(3^16) or A^(3^17) without knowing how to find their decimal expansion.

So the possibility that someone would try to use this to claim the awards with no calculate occurred to someone (maybe Landon Noll) back around 1999.

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

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.

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.

A) A publication.

B) Eternal fame.

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

And, most importantly, it may advance mathematics.

Maybe in applied math. In pure math that question would immediately reveal you as an outsider.

Yeah, it would go something like

"What are the applications of this?"

everyone gasps, stares at you

"...to the K-theory of arithmetic cobordisms."

everything goes back to normal, speaker answers the question

A professor once told me he studied pure math _because_ it had no applications.

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