Integer Factorization and Discrete Semiprimes: A Family of Seven


Somewhere between 212 and 232 lies a set of seven equally spaced numbers which are consecutive odd discrete semiprimes. They spread out in uninterrupted uniformity, all the integers congruent to 1 modulo 4 from 481 to 505. Their prime factors, a pair each, bounce around with predictable unpredictability (so far!), as do their unique nontrivial representations as the differences of two perfect squares. To me, it’s as if one layer of mathematical noisiness drops away, so that I can look at the other layers of noisiness underneath.

The Characteristic Lines on my Zone Grid corresponding to this Family of Seven are in two different sets of evenly spaced parallels, plus one orphan: Three of the lines have slope 31, three have slope 33, and 505’s line has slope 35. Their three trivial solution Zones group together in correspondingly, but their non-trivial solutions are on different Zone boundaries except for 489=3*163 and 505=5*101, both of whose rather obvious factorizations break out at the boundary for Zone 13.

Meet the Family.
The Family, Factored.

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