In my past few days’ study of the “constellations” made by lining up the gaps between the perfect squares and the Remainder Series values for discrete semiprimes (explanation is in earlier posts), I only this morning discovered something I’d missed: There is one pair of discrete semiprimes that has the exact same gaps, and the exact same constellation chart, between the two numbers. The integers in question are 69 = 13^{2}-10^{2} and 93 = 17^{2}-14^{2}. My factorization worksheet entries, annotated a little bit for clarity, appear below regarding these two, and you can see the matching calculations, constellations and gap numbers:

```
-------------
69 = 9^2-12.(*) f(x) = x^2+18x+12.
25 - 12 = 13. <-
36 - 31 = 5. <--- Differences form a
-3. <- linear series.
36 - 12 = 24. <- Successive pairs of squares
49 - 31 = 18. <- line up with f(x) values
12. <- to produce linear interval
6. <- two less than the previous one.
-> 0.
0
<====+|-=> (pos left, neg right.)
... 5 -3 .X....|..X Constellation
... 6 0 X.....X...
-------------
93 = 11^2-28. f(x) = x^2+22x+28.
49 - 28 = 21.
64 - 51 = 13.
5.
-3.
64 - 28 = 36.
81 - 51 = 30.
...
6.
-> 0.
+ 0 -
... 5 -3 .X....|..X
... 6 0 X.....X...
-------------
```

The asterisk (*) next to the ceiling-square/remainder expression of 63 is due to the fact that the square and remainder are not relatively prime, which enables a quicker solution than the search for Fermat factorization.

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