After my Tiny Eureka yesterday led to nothing new in the way of mathematical discoveries – led, in fact, to a tautology, a sort of uninformative a=a result – I took a breather. I had already scheduled myself a break after “Tiny Eureka,” and it felt strange to pull away from the study on which I had spent so much thought and calculation. I surprised myself with my mental reaction to the cessation of intense investigation, albeit for a break of such short duration: the way other daily experiences felt, other mundane thought processes proceeded. I think that came from some part of my mind not wanting to let it go, even for part of a day. Maybe a silent part of my mind wasn’t letting it go.

Why would my mind do that? There are psychological reasons, I am sure, but I also began to wonder if there were mathematical reasons. Did the fact that I have been barking up a succession of wrong trees mean that there were no right trees to sniff? My Tiny Eureka gave me, not a new revelation, but certainly a deeper understanding on my part of the behaviors of numbers I am studying. I truly am not yet ready to decide that the search for the right tree up which to bark, the right avenue of attack on integer factorization that will use what I have learned so far and that will bear fruit, is doomed to ultimate failure.

No. I’m not ready to stop. I am going to look right now at tables of values of s, r, c_{1}, and c_{2}, described in earlier posts, and study their relationships for further ways I can use them. The relationship between the primes p and q and the difference r between s^{2} and pq is not one-to-one, and thus it is not invertible. But maybe bringing s back into the picture will help with that, and so also might expressing p as s-c_{1} and q as s+c_{2}, so that there are a couple more positive integers to engage and examine for properties.

I also want to produce a Google Document of all the genuine results I have uncovered so far, however well known they may be in standard number theory, along with a description of my research process, complete with tables and tales of looking at much data.

I still hope to find something significant, and I still feel at least somewhat confident that I have put myself on a track that could lead me there, in the approaches I have taken to this study, and the ideas I have had that have become prolific and exhilarating.

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