To me, pretty mathematical properties are always exciting. They do not always lead to new discoveries of great mathematical moment, however. Right now, I’ve discovered something pretty, that may lead to other discoveries, which is pretty exciting in itself to me, whether momentous or not.

I am developing something new I call Remainder-Remainder Tables, which I will explain in detail in a post tonight, if all goes according to plan. Below is a screenshot of two such tables under construction. But first, I will share the pretty discovery that is *not* of great moment: It’s just an instance where the numbers involved and their relationships are beautiful. In the course of pursuing and assembling these tables today, I decided to look at the odd integer 101010101. In chasing down its Ceiling Square, Remainder, and Ascent, I happened on a beautiful fact about its factorization:

`101010101 = 10101`^{2}-1010^{2}.

And now, the teaser for the Remainder-Remainder Tables: Both will be filled out, perhaps even augmented, and with an explanation tonight.

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