4 ms·

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

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 12y ago
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 12y ago
  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_constant https://en.wikipedia.org/wiki/Copeland%E2%80%93Erd%C5%91s_co...
  - logicallee 12y ago
    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

logicallee 12y ago

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 12y ago
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_constant https://en.wikipedia.org/wiki/Copeland%E2%80%93Erd%C5%91s_co...
- logicallee 12y ago
  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