We haven't successfully resolved famous unsolved research problems through language models yet but one can imagine that they will solve increasingly challenging problems over time. And if it happens in the hands of a researcher rather than model's lab, one can also imagine that the researcher will take credit, so you will still have the same question.
My general sense is that for research-level mathematical tasks at least, current models fluctuate between "genuinely useful with only broad guidance from user" and "only useful after substantial detailed user guidance", with the most powerful models having a greater proportion of answers in the former category. They seem to work particularly well for questions that are so standard that their answers can basically be found in existing sources such as Wikipedia or StackOverflow; but as one moves into increasingly obscure types of questions, the success rate tapers off (though in a somewhat gradual fashion), and the more user guidance (or higher compute resources) one needs to get the LLM output to a usable form. (2/2)Show me the code. Show me your finished product.
After all, the LLM currently has no cognizance, it is unable to understand what it is saying in a meaningful way. At its best it is a P-Noid Zombie machine, right?
In my opinion anything amazing that comes from an LLM only becomes amazing when someone who was capable of recognizing the amazingness perceives it, like a rewrite of a zen koan, "If an LLM generates a new work of William Shakespeare, and nobody ever reads it, was anything of value lost?"