Can I use it? And if not: when can I use it?

State-of-the-art LLM agents do not perform calculations, they call external tools to do that.

To be fair, not all knowledge of LLM comes from training material. The other way is to provide context to instructions.

I can imagine someone someday develops a decent way for LLMs to write down their mistakes in database and some clever way to recall most relevant memories when needed.

It is literally an algorytm made to hallucinate. The fact that it outputs accurate facts is more of a side effect.