I just want to be sure that the upper bound for
in my Table can be improved to
.
Ok, I don't have time to reduce to a minimal set, but this combination is sufficient to prove it.
In the first set, no value in a chain can have one of these modular values because they contribute a factor to
that does not divide 120; in the second set, no value in a chain can have one of these modular values because they require an unattainable square.
Set 1: 64 mod 128; 36, 180 mod 216; 256 mod 512; 100, 300, 700, 900 mod 1000; 216, 1080 mod 1296; 288, 1440 mod 1728.
Set 2: 120 mod 144; 168, 264 mod 288; 270 mod 324; 120, 280 mod 400; 384 mod 512; 378, 594 mod 648; 168, 312, 408, 552 mod 720; 640 mod 768; 440, 760 mod 800; 210, 330, 390, 510, 690, 870 mod 900; 440, 680, 920, 1160 mod 1200; 384, 896 mod 1280; 378, 702, 918, 1242 mod 1620.