Can I publish a single page research paper?

Yes, if your ideas can be expressed concisely enough. For example if you were to find a counterexample to the Beal Conjecture, then you can write two paragraphs and be done. Watson and Crick's 1953 paper describing the structure of DNA was one page long, and won the Nobel Prize. Some other examples of very short papers (admittedly, some of these are jokes) are here.

My favorite short paper, which like Nash's in another answer is from the 1950s, is as follows:

I do have a hard time seeing how it would be published these days, but clearly very short papers have been acceptable in the past.

John Nash published a paper entitled Equilibrium Points in n-Person Games. The paper is one page long. But it's John Nash, and it was in 1950. It was the beginning of a whole new field (game theory), and the man was a genius. (Credit to MathOverflow.)