I get
; longer chains are disallowed for
by considering 11-smooth moduli up to 2310. (That seems like an interesting result, if I get time I'll try to get the full list of constraints that disallow this.)
I finally wrote a program to categorize the constraints and throw out duplicates (though it does not yet optimize by removing constraints that are not needed for the final result). That tells me this came from the constraints below, which should at least be possible to verify.
(Оффтоп)
Set 1 (tau does not divide n): 24, 120 mod 144; 36, 180 mod 216; 128 mod 256; 270 mod 324; 40, 120, 280, 360 mod 400; 256 mod 512; 56, 168, 280, 504, 616, 728 mod 784; 510, 690 mod 900; 100, 300, 700, 900 mod 1000; 1080 mod 1296; 288, 1440 mod 1728; 88, 264, 440, 616, 792, 1144, 1320, 1496, 1672, 1848 mod 1936; 270, 1080, 1485, 1890 mod 2025; 1024 mod 2048
Set 2 (forces impossible square): 24 mod 32; 30 mod 36; 40 mod 48; 42 mod 72; 24, 56 mod 80; 54 mod 81; 30, 70 mod 100; 24, 40, 104 mod 112; mod 144; 56, 104, 136, 152, 168 mod 176; 42, 78 mod 180; 42, 70, 182 mod 196; 110, 190 mod 200; mod 216; 30, 105, 120, 165, 195, 210 mod 225; 78, 114, 186 mod 252; mod 256; 110, 290 mod 300; 189 mod 324; 154, 210, 322 mod 392; 42, 78, 114, 258, 366 mod 396; mod 400; 189, 351 mod 405; 360 mod 432; 42, 105, 168, 210, 231, 273, 357, 399, 420 mod 441; 154, 286, 374, 418, 462 mod 484; mod 512; 270, 351, 513 mod 567; 480 mod 576; 154, 322, 406 mod 588; 250, 375 mod 625; 190, 310, 410, 590, 610, 690 mod 700; mod 784; 189, 270, 351, 513, 756 mod 891; 285, 465 mod 900; 66, 110, 330, 506, 594, 682, 814 mod 968; 322, 518, 602, 742, 798, 938 mod 980; mod 1000; 66, 165, 231, 264, 330, 429, 462, 561, 627, 660, 759, 858, 924, 957, 1056 mod 1089; 190, 210, 290, 390, 410, 510, 610, 790, 1010, 1090 mod 1100; 672 mod 1152; 70, 105, 210, 280, 420, 455, 595, 630, 665, 770, 805, 840, 910, 945, 1085, 1120, 1155, 1190 mod 1225; 756 mod 1296; 110, 506, 1034, 1166, 1298 mod 1452; 285, 465, 510, 915, 1140, 1410 mod 1575; 480, 1120 mod 1600; mod 1728; 903, 1407, 1659 mod 1764; 1375, 1750 mod 1875; mod 1936; 600, 1400 mod 2000; mod 2025; mod 2048; 406, 546, 602, 714, 798, 910, 1106, 1190, 1330, 1498, 1526, 1722, 1778, 2086, 2114 mod 2156; 462, 777, 903, 1218, 1407, 1848 mod 2205
Set 3 (impossible above limit): 0 mod 2310 > 638027694150
(I'm not sure why my initial search found longer chains are disallowed for
, when I'd expect it to be for
; that may be a bug, or I may just have fumbled.)
PS: I'm suprized by even upper bound.
Yes, me too; but maybe pushing to higher moduli would reduce it to 25.