Consider the relation ~ on $ℕ$ defined as follows:
$x$ ~ $y ⇔x * y$ is a square
Interpret x ~ y in terms of prime factors of x and y.
$ x = {P_1}^{n_1}{P_2}^{n_2}{P_3}^{n_3}{P_4}^{n_4}...{P_a}^{n_a}$
$y = {Q_1}^{m_1}{Q_2}^{m_2}{Q_3}^{m_3}{Q_4}^{m_4}...{Q_a}^{m_a}$
Where $P,Q$ are prime
Then
$ {P_1}^{n_1}{P_2}^{n_2}{P_3}^{n_3}{P_4}^{n_4}...{P_a}^{n_a}$ ~ $ {Q_1}^{m_1}{Q_2}^{m_2}{Q_3}^{m_3}{Q_4}^{m_4}...{Q_a}^{m_a}$ $⇔ ({P_1}^{n_1}{P_2}^{n_2}{P_3}^{n_3}{P_4}^{n_4}...{P_a}^{n_a})\times ({Q_1}^{m_1}{Q_2}^{m_2}{Q_3}^{m_3}{Q_4}^{m_4}...{Q_a}^{m_a})$ is a square.
This is as far as I can get. Where do I begin?