mills_constant.sf

#!/usr/bin/ruby

# Generate the Mills primes and the Mills constant.

# Definition:
#   a(0) = 2; a(n) is least prime > a(n-1)^3.

# Mills constant is M = a(n)^(3^(-n)).
# Then floor(M^(3^n)) is always prime, for n >= 0.

# OEIS sequences:
#   https://oeis.org/A051254
#   https://oeis.org/A108739

# See also:
#   https://www.youtube.com/watch?v=6ltrPVPEwfo
#   https://en.wikipedia.org/wiki/Mills%27_constant

define C = 3    # see also Legendre's conjecture

func nth_mills_prime(n) is cached {
    n<=0 ? 2 : next_prime(__FUNC__(n-1)**C)
}

func mills_constant(n) {
    nth_mills_prime(n).root(C**n)
}

var M = mills_constant(6)

say 5.of { M**(C**_) -> floor }
say 5.of { nth_mills_prime(_) }

say "\nMills constant M =~ #{M}"

__END__
[2, 11, 1361, 2521008887, 16022236204009818131831320183]
[2, 11, 1361, 2521008887, 16022236204009818131831320183]

Mills constant M =~ 2.22949477249159523572285223765557019347314904643