Daily Challenge #231 - Perfect Powers

A perfect power is a classification of positive integers:

In mathematics, a perfect power is a positive integer that can be expressed as an integer power of another positive integer. More formally, n is a perfect power if there exist natural numbers m > 1, and k > 1 such that mk = n.

Your task is to check whether a given integer is a perfect power. If it is a perfect power, return a pair m and k with mk = n as a proof. Otherwise return Nothing, Nil, null, NULL, None or your language's equivalent.

Note: For a perfect power, there might be several pairs. For example 81 = 3^4 = 9^2, so (3,4) and (9,2) are valid solutions. If a number is a perfect power, return any pair that proves it.

Examples

isPP(4) => [2,2]
isPP(9) => [3,2]
isPP(5) => None

Tests

isPP(4)
isPP(8)
isPP(14)

Good luck!

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This challenge comes from bkaes on CodeWars. Thank you to CodeWars, who has licensed redistribution of this challenge under the 2-Clause BSD License!

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Comments

[Paula Gearon]

Not necessarily the best approach, but this works in Clojure

(defn pp? [x]
  (let [logx (Math/log x)
        range-to #(range 2 (inc %))
        power? (fn [m]
                 (when-let [k (some
                                #(if (= x (long (Math/pow m %))) %)
                                (range-to (/ logx (Math/log m))))]
                   [m k]))]
    (some power? (range-to (Math/sqrt x)))))

[Vidit Sarkar]

Python solution,

import math

def isPP(number : int) -> tuple:
    if(number > 3):
        for i in range(2, int(math.sqrt(number))+1):
            power = round(math.log(number)/math.log(i), 4)
            if power.is_integer():
                return (i, int(power))

Output,

print(isPP(4)) # output -> (2, 2)
print(isPP(9)) # output -> (3, 2)
print(isPP(5)) # output -> None
print(isPP(4)) # output -> (2, 2)
print(isPP(8)) # output -> (2, 3)
print(isPP(14)) # output -> None
print(isPP(184528125)) # output -> (45, 5)

[Matt Ellen-Tsivintzeli]

Where you say mk = n do you mean "m to the power of k" or "m times k"? It looks like m times k, but your challenge talks about powers.