Successful research mathematicians do consistently good work on many problems. While there may be one or two highlights that really boost a career, few people are one-hit wonders that stumble upon a solvable problem.
Most are one hit wonders. Most research mathematicians end up having at most 1 or 2 Ph.D. students. Most publish minor extensions of known results. The greats have an insight and end up doing a lot in that one minor area. Very few people have more than a couple of truly remarkable theorems.