如果存在 b 和 e n 是一个完美的幂. em>，其中 b ^ e = n .例如216 = 6 ^ 3 = 2 ^ 3 * 3 ^ 3是一个完美的幂，但72 = 2 ^ 3 * 3 ^ 2不是一个完美的幂.

A number n is a perfect power if there exists a b and e for which b^e = n. For instance 216 = 6^3 = 2^3 * 3^3 is a perfect power, but 72 = 2^3 * 3^2 is not.

确定一个数字是否为完美幂的诀窍是要知道，如果该数字是一个完美幂，则指数 e 必须小于log2 n ，因为如果 e 更大，那么2 ^ e 将大于 n .此外，仅需测试素数 e ，因为如果数字是复合指数的完美幂，那么它也是复合成分的质数的完美幂；例如2 ^ 15 = 32768 = 32 ^ 3 = 8 ^ 5是理想的立方根，也是理想的第五根.

The trick to determining if a number is a perfect power is to know that, if the number is a perfect power, then the exponent e must be less than log2 n, because if e is greater then 2^e will be greater than n. Further, it is only necessary to test prime es, because if a number is a perfect power to a composite exponent it will also be a perfect power to the prime factors of the composite component; for instance, 2^15 = 32768 = 32^3 = 8^5 is a perfect cube root and also a perfect fifth root.

下面显示的函数isPerfectPower通过使用牛顿方法首先计算整数根，然后计算结果是否等于 n n 的素数./em>.辅助函数primes通过Eratosthenes筛子计算素数列表，iroot通过牛顿方法计算整数 k 根，而ilog计算整数对数为 b 通过二进制搜索.

The function isPerfectPower shown below tests each prime less than log2 n by first computing the integer root using Newton's method, then powering the result to check if it is equal to n. Auxiliary function primes compute a list of prime numbers by the Sieve of Eratosthenes, iroot computes the integer kth-root by Newton's method, and ilog computes the integer logarithm to base b by binary search.

def primes(n): # sieve of eratosthenes
    i, p, ps, m = 0, 3, [2], n // 2
    sieve = [True] * m
    while p <= n:
        if sieve[i]:
            ps.append(p)
            for j in range((p*p-3)/2, m, p):
                sieve[j] = False
        i, p = i+1, p+2
    return ps

def iroot(k, n): # assume n > 0
    u, s, k1 = n, n+1, k-1
    while u < s:
        s = u
        u = (k1 * u + n // u ** k1) // k
    return s

def ilog(b, n): # max e where b**e <= n
    lo, blo, hi, bhi = 0, 1, 1, b
    while bhi < n:
        lo, blo, hi, bhi = hi, bhi, hi+hi, bhi*bhi
    while 1 < (hi - lo):
        mid = (lo + hi) // 2
        bmid = blo * pow(b, (mid - lo))
        if n < bmid: hi, bhi = mid, bmid
        elif bmid < n: lo, blo = mid, bmid
        else: return mid
    if bhi == n: return hi
    return lo

def isPerfectPower(n): # x if n == x ** y, or False
    for p in primes(ilog(2,n)):
        x = iroot(p, n)
        if pow(x, p) == n: return x
    return False

在我的博客中，有关于完美幂谓词的进一步讨论. >.