Should I publish a research paper on my blog if I work by myself?

That depends on your intentions for it. If you hope to score a journal article, I suggest not publishing it on the blog. Most journals will decline a manuscript if it was previously published (and I suspect they would count a blog as “published”), and many require that you assign them copyright and then restrict your ability to disseminate free copies (which would be a problem if you had already posted it).
Assuming you want to get a journal pub out of it, you can (if you wish) do a blog post giving a sketch of what you did, without details. (“In a hopefully forthcoming paper, I demonstrate that P = NP in an alternate universe where computational speed and memory automatically grow in concert with problem size. The paper includes computational results from a dream I had last night, and an extensive bibliography of similar research by the ancient Mayans.”) Then, if and when it is published, you can blog that (perhaps with a link supplied by the journal publisher)

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