How can I get a PhD position to work on solving a particular mathematics problem that I have formulated independently?
Unfortunately, I expect it will be difficult to find a supervisor who will take you on under these circumstances.  It's not impossible, but it will require finding exactly the right match, and that will require a lot of luck.

Here are some of the issues that will worry potential advisors:


What if the problem is too easy, and turns out not to be substantial enough for a Ph.D. thesis?  Will you drop out from graduate school in disgust?  Will you angrily insist that you signed up to write your dissertation on this topic and now they have to let you graduate?
What if the problem is too hard, and you spend several years in graduate school with no hint of progress or even the potential for future progress?  When your advisor suggests changing course, will you listen?  Or will you keep banging your head against this problem until they kick you out for lack of progress?
What if the problem doesn't end up being nearly as interesting or important as you hope?  Will you accept this outcome, or will you turn into a crackpot who insists that the rest of the research community is just wrong?
Are you too emotionally invested in this problem to judge whether you are making progress or how interesting it is?  How painful or awkward would it be to work with you on this topic?
Are you interested in learning how to do anything else besides work on this specific problem?  The purpose of a Ph.D. program is to train researchers with some breadth, not just to solve one specific problem, so you need to maintain some perspective and longer-term goals than any single problem.


It's possible that none of these issues is an actual problem, but few people will agree to supervise this thesis unless they have enough knowledge of both you and your problem to feel reassured.  Unfortunately, the bar for "enough knowledge" is probably pretty high.  I wouldn't do it unless I knew you relatively well and had spent some time thinking seriously about the problem.  This is not going to happen during the ordinary graduate admissions process.

Instead, I think you're probably best off not worrying too much about graduate admissions, and instead just focusing on convincing people that this is an important problem.  If you can make a real intellectual connection with a professor based on this problem, then you can explore whether it might be a suitable topic for a Ph.D. thesis.  Otherwise, applying to graduate school for the sole purpose of working on this problem is likely to be fruitless.

(To be clear, I might be wrong about this, but it's my impression based on the limited information in your question.)