It sounds like you have a singular focus on achieving the following two goals:
Getting into a fully funded, high quality pure math PhD program.
Using your position as a funded PhD student to solve a particular math problem you have formulated, and nothing else.
Let's start by thinking how you can achieve the first of these two goals. The thing to keep in mind is that PhD programs decide whom to admit based on the applicant's potential to successfully complete a PhD. So, the extent to which your plan formulated in item 2 above will help you get into the PhD program depends precisely on what it says about your potential to complete the PhD.
Unfortunately, by stating that you are uninterested in working on any other math problem, I think you are likely to hinder your application, and make it less likely that you are admitted, by showing that you have less potential to successfully complete a PhD than other students with more normal plans. The reason for this is that you are taking an approach to math research that any experienced mathematician knows is a bad one, which is to "put all your eggs in one basket". The fact is, most research problems that mathematicians work on end up leading nowhere, and for this reason experienced mathematicians know that in research one must be open minded and try to work on many problems to have a good chance of success, and that even when working on a specific problem one must be flexible and willing to change one's goal as one acquires a more nuanced understanding of the problem.
Likewise, if you dedicate yourself to solving your problem and doing nothing else, it is overwhelmingly more likely that one of the following things will happen rather than your dream scenario of solving it and getting a PhD based on that solution:
The problem will end up being too difficult and you will not succeed in solving it.
You will discover that the problem is known and is solved in some obscure paper from the 1950s, or is an easy corollary of a well-known result.
You will discover that the problem is easy and has a trivial solution, and moreover the application you had in mind will turn out to be not as useful as you thought, making the result unpublishable.
The problem will end up being at a good difficulty level. You will solve it after a year or two of work, only to discover that for subtle reasons you did not appreciate beforehand, the application you had in mind is not as useful as you thought, making the result publishable but not very interesting, and not enough to get a PhD for or to make you feel that the whole endeavor was worth the effort you put into it.
To summarize, your attitude as it currently stands positions you as a difficult person with unrealistic goals and expectations about what it means to be in a PhD program -- not a good image for someone trying to get into such a program. If you really want to achieve goal number 1, let alone goal number 2, you need to think about how you can present yourself as a more attractive candidate. As Anonymous Mathematician explained, that means being more open minded and expressing curiosity about getting a broad exposure to many areas of math and many research problems, and not just showing a singular desire to perform one feat you have set your mind on and which no one else is yet convinced is either worth doing or feasible to perform.
You also need to get rid of the mindset (as expressed in your comment "You know how difficult it is to come up with a provable mathematics statement thats not trivial and absent in the literature?") that by formulating a new math problem you have done something amazing that shows some special level of talent. You may have done something amazing, and you may be really talented, and the fact that you came up with a reasonable-sounding open problem does say something mildly nice about you, but frankly, open problems are a dime a dozen (one can easily find hundreds of them on MathOverflow). You can mention the problem in your Statement of Purpose, and perhaps some people on admissions committees will be impressed by it, but your unhealthy focus on the problem at the moment is much more likely to hurt your application than it is to help it.