# Prikry forcing and Cohen generic

Let $\kappa$ be a measurable cardinal and let $\mathcal{U}$ be a normal measure on $\kappa$. Let $\mathbb{P}$ be the standard Prikry forcing using $\mathcal{U}$. Let $\mathbb{Q} = \text{Add}(\kappa, 1)$ be Cohen forcing for adding a new subset to $\kappa$ using partial functions from $\kappa$ to $2$ of size ${<}\kappa$.

Question: Is there a projection from $\mathbb{P}$ to $\mathbb{Q}$?

Theorem. Suppose $V$ satisfies $GCH$ and $\kappa$ is a measurable cardinal. Then in a cofinality preserving generic extension, there exists a $\kappa$-complete ultrafilter $U$ on $\kappa$ such that Prikry forcing with $U$ adds a Cohen subset of $\kappa$ over $V$.