tower_of_powers_inverse.sf

#!/usr/bin/ruby

# Daniel "Trizen" Șuteu
# Date: 05 June 2020
# https://github.com/trizen

# Find a real solution to x^x^x^...^x = n for a tower of k powers.

func tower_of_powers_inverse(n, k) {  # solution for x^x^...^x = n
    bsearch_inverse(n, {|v|
        k.of(v).reduce{|a,b|
            b**a
        } <~> n
    })
}

func nested_powers_inverse(n, k) {   # solution for ((...)^x)^x = n
    bsearch_inverse(n, {|v|
        k.of(v).reduce{|a,b|
            a**b
        } <~> n
    })
}

# Find x such that x^(x^(x^x)) = 2
# x = 1.4466014324298641745973339875976614806873210422...
with (tower_of_powers_inverse(2, 4)) {|x|
    say x
    assert(x**(x**(x**x)) =~= 2)
}

# Find x such that ((x^x)^x)^x = 2
# x = 1.3367097349494955658483777999091512580783714292...
# Equivalently: x^(x^3) = 2
with (nested_powers_inverse(2, 4)) {|x|
    say x
    assert(((x**x)**x)**x =~= 2)
}