Equivalence of The Two Definitions
If m = a2b3, then every prime in the prime factorization of a appears in the prime factorization of m with an exponent of at least two, and every prime in the prime factorization of b appears in the prime factorization of m with an exponent of at least three; therefore, m is powerful.
In the other direction, suppose that m is powerful, with prime factorization
where each αi ≥ 2. Define γi to be three if αi is odd, and zero otherwise, and define βi = αi - γi. Then, all values βi are nonnegative even integers, and all values γi are either zero or three, so
supplies the desired representation of m as a product of a square and a cube.
Informally, given the prime factorization of m, take b to be the product of the prime factors of m that have an odd exponent (if there are none, then take b to be 1). Because m is powerful, each prime factor with an odd exponent has an exponent that is at least 3, so m/b3 is an integer. In addition, each prime factor of m/b3 has an even exponent, so m/b3 is a perfect square, so call this a2; then m = a2b3. For example:
The representation m = a2b3 calculated in this way has the property that b is squarefree, and is uniquely defined by this property.
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