2048th post!

[Yoni]

Mathematical discussion to make this post fit for my forum.

[04:05:23] <BinaryChat> 2^11 posts is really worth noting?
[04:05:53] <Yoni[vL]> more worth than 2*(2*5)^3
[04:06:17] <Yoni[vL]> it is the first number that isn't 11-free
[04:06:46] <BinaryChat> Surely you could have used a more obscure way to say 2000.
[04:07:22] <Yoni[vL]> where x being n-free means in the prime factorization of x, no prime appears n or more times
[04:07:47] <Yoni[vL]> for example, squarefree (2-free) means it doesn't contain squares, i.e. prime factorization generates distinct primes
[04:10:26] <Yoni[vL]> it is known that the asymptotic density of n-free numbers is 1/zeta(n)
[04:11:12] <[vL]Kp> zeta(n) = ?
[04:11:23] <Yoni[vL]> the Riemann Zeta function
[04:11:27] <Yoni[vL]> the most important function in mathematics
[04:11:37] <Yoni[vL]> shelves full of books have been written about this function
[04:11:51] <[vL]Kp> None of the shelves are within my reach atm :p
[04:12:10] <Yoni[vL]> for an positive real s, i believe zeta(s) can be given by the formula...
[04:12:30] <Yoni[vL]> zeta(s) = sum(n=1,infinity) n^-s
[04:12:46] <[vL]Kp> Well, why didn't you say so in the first place?
[04:12:47] <Yoni[vL]> i.e. zeta(2) = 1/1^2 + 1/2^2 + 1/3^2 + 1/4^2 + ...
[04:12:51] <[vL]Kp> yeah
[04:13:02] <Yoni[vL]> = pi^2 / 6 (oh the beauty of calculus)
[04:13:51] <Yoni[vL]> for an even positive integer n, it is known that zeta(n) = pi^n / (some whole number)
[04:14:10] <Yoni[vL]> for an odd integer n, zeta(n) / pi^n doesn't give a rational number or a meaningful number
[04:15:01] <Yoni[vL]> anyway, the density of 11-free numbers is 1/zeta(11), and the density of 12-free numbers is 1/zeta(12)
[04:15:23] <Yoni[vL]> meaning the density of "just-12-free numbers" has to be 1/zeta(12) - 1/zeta(11)
[04:15:27] <Yoni[vL]> let's see what that number is
[04:16:12] <Yoni[vL]> hmm, i was wrong, it's not pi^n / (some whole number), it's pi^n * (some rational number). oh well, same thing
[04:16:36] <Yoni[vL]> anyway, up to 10 digits, that is
[04:16:36] <Yoni[vL]> 0.0002479184928
[04:16:44] <Yoni[vL]> let's multiply it by a 1000000
[04:16:50] <Yoni[vL]> 247.9184928
[04:16:56] <Yoni[vL]> meaning, BinaryChat
[04:17:07] <Yoni[vL]> that less than a million, only about 248 numbers are as special as 2048.
[04:17:37] <Yoni[vL]> what's special about 2000? it is 4-free but not 3-free.
[04:17:59] <Yoni[vL]> 1/zeta(4) - 1/zeta(3) = 0.09203103034
[04:18:12] <Yoni[vL]> 1000000/zeta(4) - 1000000/zeta(3) = 92031.03034
[04:18:28] <Yoni[vL]> less than a million, there are 248 numbers as special as 2048, but over 90,000 numbers as "special" as 2000