is_perfect_power_fast.sf

#!/usr/bin/ruby

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

# A fast pretest for checking if a number `n` is a perfect power.
#      (i.e. can be expressed as: n = a^k with k > 1)

# Algorithm:
#   1. Let `n` be the number to be tested.
#   2. Let `k` be the primorial of `B`.
#   3. Compute the greatest common divisor: `g = gcd(n,k)`.
#   4. If `g` is greater than `1`, then `n = r * g^e`, for some `e >= 1`:
#       * if `r = 1` and `e = 1`, then `n` is not a perfect power.
#       * if `r = 1` and `e > 1`, then `n` is a perfect power.
#       * if `r` is prime, then `n` is not a perfect power. (optional)
#       * Factorize `g` into primes `p_k` and compute multiplicity `e_k` in `n`:
#           * if `gcd(e_k) = 1`, then `n` is not a perfect power.
#           * if `gcd(e_k) = b`, with `b >= 2` and `n = floor(n^(1/b))^b` , then `n` is  a perfect power.
#   5. If this step is reached, then `n` is probably a perfect power.

func is_perfect_power_pretest(n,k) {

    var g = gcd(n,k)

    if (g > 1) {
        var e = valuation(n,g)
        var r = (n / g**e)

        if ((r == 1) && (e == 1)) {
            return false
        }

        if ((r == 1) && (e > 1)) {
            return true
        }

        var f = g.factor.map {|p|
            [p, n.valuation(p)]
        }

        var b = f.map { .tail }.gcd

        if (b == 1) {
            return false
        }

        var t = (n / f.prod_2d {|p,e| p**e })

        if (t.is_prime) {
            return false
        }
    }

    if (n.is_prime) {
        return false
    }

    return true
}

func is_perfect_power(n) {

    for k in (n.ilog2 `downto` 2) {
        if (n.iroot(k)**k == n) {
            return k
        }
    }

    return 1
}


say is_perfect_power(1234**14)     #=> 14
say is_perfect_power(27*49)        #=> 1

var k = 29.primorial
say is_perfect_power_pretest(1234**14, k)       #=> true
say is_perfect_power_pretest(27*49, k)          #=> false

var a = (2..1000 -> grep { is_perfect_power(_) > 1 })
var b = (2..1000 -> grep { is_perfect_power_pretest(_, k) })

assert_eq(a, b)