Math Paper and Grad School Application I am applying for pure Math Ph.D programs this year. 

I did some independent research in centralizer of matrices, and I developed an algorithm to generate the explicit k-basis for the space of centralizers. I guess this problem is not solved since I cannot find any papers on this topic, and a similar question asked on Math Stack Overflow has no complete solutions.

I asked my professor (one of my recommendation letter writer) to take a look, but this is irrelevant to his fields of studies and he said that he might have no time to read this. 

I want this research experience to strengthen my application. So, how should I deal with my paper? Shall I just mention it in my resume and personal statement or should I post the paper on arXiv? I am not sure what is appropriate.
If your goal is to have a way to include this in your graduate applications, you can certainly talk about this project in a statement of interest.  Search this site for "statement of interest" and you will see many have no idea what to put there.  You have something concrete to talk about!  Did this experience make you want to study more algebra, or are you ready to switch to analysis?  Do you want to look at applications, or more theory, of matrices over finite fields?

It is a slight disappointment that you did not do this as an independent study, which is what someone my age would have done.  On the other hand, you did get feedback on MathOverflow.  Graduate admissions committees can take into account interactions like this.  I would say this shows independence and a willingness to learn by whatever means available.  You cannot say that in a statement of interest, but you can talk about how you enjoy having Denis Serre answer a question of yours.  (Name-dropping is best done shamelessly.)