What have I been jumping down into this week?

After making and sharing parts of some studies into how odd positive integers with fixed remainders from their Ceiling Squares distribute themselves, and constructing companion lists of factorizations and Ascents for differences of even/odd and odd/even pairs of perfect squares relatively prime to each other without being consecutive, I’ve taken a detour.

In particular, it occurred to me to examine quadratic residues of n^{2} for choices of n considered as numerical bases: In other words, I’m looking at the last two possible digits of perfect squares in base n, and examining the proportion of these possibilities among the n^{2} possible two-digit endings of numbers in base n. For decimal integers, there are only 22 such ways a perfect square can end, or 11/50 of the possible two-digit endings. I’ve not figured out a formula for the proportion for a given base, but that will be a profitable exercise for me or others to attempt.

Where it fits into factorization is that one can use it to winnow down the possibilities for s^{2}-t^{2} that will produce a given tens-and-ones digit. One can also go into other bases, or figure out a similar table of s, t pair possibilities for more than just the last two digits. (Base n^{i} for i greater than two.)

I also had fun making square tables for small bases, including decimal, hexadecimal, and smaller bases:

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