Is it acceptable to have a research paper with no references?
Though I don't think there's any hard rule against having a paper with no reference, it seems pretty weird. Note that references are not only for citing other people's results (theorems, algorithms, etc.) which you have used, but more broadly to recognize other's scientific contribution. For example:


Has the problem never been discussed before? Who first realized it was a problem, stated it, formalized it?
You probably put the problem into the broader context of your field. And if you don't, you probably should. This sure requires citations on recent work on related problems, even if nothing was ever done on the one problem you're addressing.
For example, is your problem a specific case of another problem, or does it have generalization?
What are the consequences of your results/findings? They probably have some impact on other related problems, or practical consequences on real-life issues.
Didn't you or your co-authors ever do any prior work on this issue?




Finally, let's see it another way: you have solved a problem, that no one else has solved, worked on, or more generally discussed. And you did so using only elementary techniques, which have been known for so long that they do not require citation. Stated like that, it may sound like either you're a genius opening an entire field of mathematics, or you're working on a useless problem that nobody cares about. You probably don't want the reviewer to be thinking that way!