整数幂次等价性证明问询：36次完全幂的充要条件验证

Hey there! Let's break down this equivalence proof step by step—prime factorization is going to be our best tool here, since it's perfect for analyzing integer powers.

必要性：若 (a = n^6 = k^9 = m^4)，则 (a) 是36次完全幂 #

First, let's use the prime factorization of (a). Any positive integer can be written as a product of primes raised to exponents, so let's define:
a = p₁^e₁ * p₂^e₂ * ... * p_t^e_t
where each (p_i) is a prime number, and each (e_i) is a positive integer.

Since (a = n^6), every exponent (e_i) must be a multiple of 6—when you raise a number to the 6th power, all its prime exponents get multiplied by 6, so reversing that means each (e_i) has to be divisible by 6. In shorthand, that's 6 | e_i for every (i).

By the same logic:

• (a = k^9) implies 9 | e_i for every (i)
• (a = m^4) implies 4 | e_i for every (i)

Now we need the smallest number that's a multiple of 6, 9, and 4—this is their least common multiple (LCM). Let's calculate it:

• 6 factors into (2×3)
• 9 factors into (3²)
• 4 factors into (2²)
  The LCM takes the highest power of each prime present: (2²×3² = 36). So every (e_i) is a multiple of 36, meaning we can write (e_i = 36×f_i) for some integer (f_i).

Substitute this back into the prime factorization:
a = p₁^(36f₁) * p₂^(36f₂) * ... * p_t^(36f_t) = (p₁^f₁ * p₂^f₂ * ... * p_t^f_t)^36
If we let (s = p₁^f₁ * p₂^f₂ * ... * p_t^f_t) (which is an integer, since it's a product of primes raised to integer exponents), then (a = s^36)—so (a) is a perfect 36th power.

充分性：若 (a) 是36次完全幂，则存在整数 (n,k,m) 使得 (a = n^6 = k^9 = m^4) #

Suppose (a = s^36) for some integer (s). We can easily construct the required integers by dividing the exponent 36 by the powers we need:

• Let (n = s^(36/6) = s^6). Then (n^6 = (s⁶⁾6 = s^(6×6) = s^36 = a)
• Let (k = s^(36/9) = s^4). Then (k^9 = (s⁴⁾9 = s^(4×9) = s^36 = a)
• Let (m = s^(36/4) = s^9). Then (m^4 = (s⁹⁾4 = s^(9×4) = s^36 = a)

All of (n, k, m) are integers because 36 is evenly divisible by 6, 9, and 4. This satisfies the original condition perfectly.

Putting both directions together, we've proven the equivalence: (a = n^6 = k^9 = m^4) if and only if (a) is a perfect 36th power.

备注：内容来源于stack exchange，提问作者user1202451