You cannot patent a prompt.
Anyone who attempts to do so will either need to be secretive and bribey to pull it off and will then also have to hope that no one ever finds out because if they did, then in any legal proceedings that come due to said patent, (which is the entire point of a patent), the outcome will be that their patent is invalidated by the court and the case thrown out.
The point of a patent is to make it so that, while the patent lasts, no one else can use said patent without licensing it from you, and if they do so without the license they open themselves up to civil penalties that can exceed any potential profits from the illicit licensing.
1: Prompts have no value in and of themselves.
2: The ownership of raw AI generated materials is in a grey area. It might be public domain, it might be owned by the person who prompted it, we don't have clear hard guidelines on that.
3: AI is trained on a dataset of billions of pieces of other peoples work. Any novelty (novelty is a requirement for a patent) that comes from AI is at best an accident of the same variety as the colors produced by spilling paint, where it has taken a slice from one person and a slice from another person and mixed them together.
So, all in all, AI prompts and their output are almost certainly not patentable, at least as long as one sane person is in the pipeline somewhere.