Conjecture:
$\forall $ $p_{n}$, $p_{n+1} \in \mathbb{P}$, $\:$ $p^2_{n+1} = p^2_{n} +\omega_{n} p_{n} + \phi_{n} : \phi_{n} , \omega_{n} \in \mathbb{N} $ and $ \phi_{n} < p_{n}$, $\:$ $\Rightarrow$ $\:$$\exists$ atleast $ \omega_{n}$ prime numbers in the interval $(p^2_{n},p^2_{n+1})$
How would you prove this?