## Integer Factorization: a Small Observation on Remainder Series Constellations

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 = 132-102 and 93 = 172-142. 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.