Editing Powerful number (section)

== Equivalence of the two definitions ==

If ''m'' = ''a''<sup>2</sup>''b''<sup>3</sup>, 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
:<math>m = \prod p_i^{\alpha_i},</math>
where each ''α''<sub>''i''</sub> ≥ 2.  Define ''γ''<sub>''i''</sub> to be three if ''α''<sub>''i''</sub> is odd, and zero otherwise, and define ''β''<sub>''i''</sub> = ''α''<sub>''i''</sub> − ''γ''<sub>''i''</sub>.  Then, all values ''β''<sub>''i''</sub> are nonnegative even integers, and all values γ<sub>i</sub> are either zero or three, so

:<math>m = \left(\prod p_i^{\beta_i}\right)\left(\prod p_i^{\gamma_i}\right) = \left(\prod p_i^{\beta_i/2} \right)^2 \left( \prod p_i^{\gamma_i/3}\right)^3</math>

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''/''b''<sup>3</sup> is an integer. In addition, each prime factor of ''m''/''b''<sup>3</sup> has an even exponent, so ''m''/''b''<sup>3</sup>  is a perfect square, so call this ''a''<sup>2</sup>; then ''m'' = ''a''<sup>2</sup>''b''<sup>3</sup>. For example:

:<math>m = 21600 = 2^5 \times 3^3 \times 5^2 \, ,</math>
:<math>b = 2 \times 3 = 6 \, ,</math>
:<math>a = \sqrt{\frac{m}{b^3}} = \sqrt{2^2 \times 5^2} = 10 \, ,</math>
:<math>m = a^2b^3 = 10^2 \times 6^3 \, .</math>

The representation ''m'' = ''a''<sup>2</sup>''b''<sup>3</sup> calculated in this way has the property that ''b'' is [[Square-free integer|squarefree]], and is uniquely defined by this property.