Can successful mathematicians become top programmers if they want to?

Almost anyone can learn to program. Someone who understands math probably has some advantages. As for “becoming” a “top programmer,” who can say. What does “top programmer” mean? A programmer who makes good money? A programmer with a recognized name in the business? No one “becomes” a top programmer because programming describes something a person does, not something they are. Success in any field depends on talent, skill, persistence, and a lot of luck.

Of course a “successful mathematician” (whatever success means in that context) can learn to program and may get to the top of the profession (again, whatever that means, which is not really clear to me). Anyone might do that, mathematician or not. No one can predict who, mathematician or otherwise, will actually succeed at programming.

Stephen Wolfram might serve as a role model for a successful mathematician who also contributed useful high-quality software, and made a lot of money doing it.

Some successful mathematicians can become top programmers if they want to.

If the question is can all of them - no. The skills required overlap but they are are different.

yes