I am currently examining my table of odd composite numbers that I have used for the 3D graph and the charts I shared a day or two ago, sorted according to remainder, and at the distribution and factors of the odd composite numbers for each such remainder. This was what led me to notice the remainders all being congruent to 0 or 1 modulo 4 when the odd/even polarity of the ceiling square was adjusted (by adding 1, if necessary) to match that of (m+1)/2. When the remainders are perfect squares, the composite integers with each of those remainders easily characterize themselves with linearly-ascending factor pairs. When they are not perfect squares, their distribution needs more insight to find a useful pattern.

I also started looking at the property that a^{2}-b^{2}=c^{2}-d^{2} if and only if a^{2}-c^{2}=b^{2}-d^{2} and what that means for a discrete semiprime m=pq:

```
((m+1)/2)^2-((m-1)/2)^2=((p+q)/2)^2-((p-q)/2)^2,
factoring m two ways.
((m+1)/2)^2-((p+q)/2)^2=((m-1)/2)^2-((p-q)/2)^2,
resulting in (m+p+q+1)(m-p-q+1)=(m+p-q-1)(m-p+q-1).
```

I believe I’ve seen something like this result before in mathematical references (Brahmagupta?), but it was gratifying to have it pop out from my own calculations. And the resulting companion pairs of factored integers, one of which commonly seems to have 48 as a factor, are interesting and, I believe, deserve further study.

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