I've recently submitted a paper for review for a reputed math journal. The main paper is just 2 pages (everything else being in appendix), where I setup a new problem, and the central result I had left unproven. (I said the proof is yet to be done). I did not send this paper in a hurry, I had spent a good amount of time and best of my abilities to prove it, but couldn't. The paper cleared editorial screening (after being there for a few weeks) and went to "Under Review". What could be the reason to go to "under review", when there is no proof that has to be reviewed. I'd like to understand, what are all the things that are "reviewed" for a math paper that is "under review"?
There's a good chance the journal is getting confirmation that it is indeed a new problem. The fact that it's new to you does not mean it's actually new - perhaps you've simply not seen the paper(s) that stated and maybe even solved the problem.
The journal could also be confirming if the problem is actually interesting. It's not so difficult to come up with a new problem, but coming up with an interesting new problem would be something else.
Why woudln't they review your paper? All reputable journals review all papers that they consider publishing..
The fact that there's no proof to check doesn't mean there's nothing to check. Arguably, in a mathematics paper, it means there's more to check: why should they accept your paper that doesn't prove anything? The fact that you put everything in an appendix doesn't mean it doesn't get reviewed. The journal will be publishing that appendix, so they want to know that it's OK. (Otherwise, everybody's next paper would be "Abstract: [blah blah] Introduction: See appendix." and publishing just got a whole lot easier.)