GilGil wrote: October 29th, 2026, 12:10 am
Period = 30 × (2¹²⁷ − 1) = 5,104,235,503,814,076,951,950,619,111,476,523,171,810 :
Code: Select all
x = 559, y = 533, rule = B3/S23
13$292bo$291bobo$290bob2o15b2o$284b2o3b2ob2o14bobo$284b2o4bob2o13bo6b
2o$291bobo13bo2bo2bo2bob2o$292bo5bo8bo6b2o2b2o$298bobo7bobo$298b2o9b2o
5$249b2o40bo$249bo2bo37bo$253bo9bobo24b3o$237bo15bo7bo3bo$234b4o15bo7b
o$225bo7b4o12bo2bo7bo4bo8b2o$224bobo6bo2bo12b2o10bo12b2o$223bo3b2o4b4o
24bo3bo$212b2o9bo3b2o5b4o14bobo8bobo17bo$212b2o9bo3b2o8bo14b2o29bobo$
224bobo26bo24b2o3b2o$225bo51bo5bo$234bobo39bo12b2o$235b2o29b2o8bo7bo4b
2o$235bo30b2o8bo8b2o5b2o$246bo30bo4b2o2bo5b3o5b2o$244b2o32b2ob5o6b2o6b
2o$245b2o37bo4b2o$289b2o$242bo13bo2bo$243b2o15bo11b4o$242b2o12bo3bo10b
o3bo$246b3o8b4o14bo$248bo22bo2bo$237bo9bo$236bobo38b2o$224b2o10b2obo8b
obo26bobo20b2o$224b2o10b2ob2o6bo2bo26bo22bo$236b2obo6b2o10b2o28bo9bobo
$236bobo5b2o3bo8b2o26b4o8b2o$237bo8b2o5b2o21bobo6bobob2o$247bo2bo4bo
20bo3bo3bo2bob3o$248bobo29bo4bobob2o$276bo4bo4b4o$280bo7bo$268b2o6bo3b
o$267bobo6bobo$267bo$266b2o23$231bo$231bobo$231b2o162$538b2o$538b2o3$
54bo$54b3o$57bo$56b2o480b3o$537b2ob2o$537b2ob2o$537b5o$536b2o3b2o3$59b
o$58b3o$57b5o$56b2o3b2o2$541b2o$541bo$58b3o481b3o$58b3o483bo3$59b2o$
59b2o224$310bobo$310bo2bo$313b2o6b2o$311bo3b2o4b2o$313b2o$302b2o6bo2bo
$301bobo6bobo$301bo$300b2o!
To compare with this other loop of similar size, whose period is much shorter :
Code: Select all
x = 641, y = 621, rule = B3/S23
50$317bo5b2o$315bo3bo3b3o$319bo5b2obo8bo$314bo5bo4bo2bo6bobo$314b2o9b
2obo5bobo$323b3o7bo2bo11b2o$323b2o9bobo11b2o$335bobo$327bo9bo$325b2o$
326b2o6$318bobo$318b2o$319bo4$312bo$311bobo$256bobo47b2o3bo$256bo3bo
45b2o5b2o$240bo19bo42b2o8bobo4bo$238b4o14bo4bo4b2o27b2o5b3o15bobo$237b
obob2o4bo12bo5b2o27b2o6b2o6bo11b2o$232b2o2bo2bob3o5b2obo3bo3bo45b2o2b
3o10b2o4b2o$232b2o3bobob2o5b4obo2bobo47b2o2bo2b2o8b2o4b2o$238b4o6b2o2b
2o57b4o5bobo$240bo71bo7bo$274b2o28b2o$260b2o11b4o13b2o11b4o$258b2ob2o
10b2ob2o10b2ob2o10b2ob2o$258b4o13b2o11b4o$259b2o28b2o14bo$247b2o56b2o$
246bobo55b2o$248bo57bo22b2o$329bo$256b2o60b2o7bobo$256bobo59bo2bo5b2o$
256bo51bo3b3o7bo$228b2o77b5o3bo6bo$226bo3bo5bo2b3o40b2o22b2ob2o3bo7bo$
220b2o3bo5bo9bo3bo36bo22b3ob2o3bo3bo2bo$220b2o2b2obo3bo8bo4bobo16b2o8b
o5bobo23b2ob4o5b2o$225bo5bo16b2o14bo2bo6bobo3b2o15b2o8b4o$226bo3bo17b
2o9b2o2bo3bo7bobo18bobo9bo$228b2o18b2o8b3o2b2o2bo7bo2bo17bo$245bobo7bo
b2o16bobo17b2o$245bo9bo2bo8b3o4bobo$255bob2o15bo$258b3o$259b2o184$570b
2o$570b2o3$82bo$82b3o487b2o$85bo$84b2o3$568b2o3b2o$569b5o$569b2ob2o$
569b2ob2o$86b3o481b3o$85bo3bo$84bo5bo$84b2obob2o2$573b2o$87bo485bo$86b
obo485b3o$86bobo487bo$87bo2$87b2o$87b2o51$159bo$160bo$158b3o173$339b2o
$337bo3bo$336bo5bo8b2o$335b2obo3bo8b2o$336bo5bo$332b2o3bo3bo$331bobo5b
2o$331bo$330b2o!
Période : 30 x (128²-1) = 491,490.
Here are a few more with maximal periods:
p(30 × (2¹⁵ − 1)) = 983,010:
Code: Select all
x = 147, y = 104, rule = B3/S23
15$106b2o$105bobo$104b3o12bo$95b2o6b3o8b2o3b4o$95b2o7b3o8b2o3b4o$105bo
bo2b2obob2o3bo2bo5b2o$106b2o2b2obobo4b4o5b2o$112b2obo3b4o$119bo3$105bo
$53b2o48b2o$51bo2bo49b2o$50bo15bobo$38bo11bo10b3o2bo3bo55b2o$38b4o8bo
19bo55b2o$22bo16b4o8bo2bo2b2o7bo4bo$21bobo5b2o8bo2bo10b2o2bo2bobo7bo$
19b2o3bo14b4o15b3o5bo3bo3b2o20bobo$14b2o3b2o3bo4bobob2o3b4o24bobo5bobo
19b2o$14b2o3b2o3bo5b2o3bo2bo37bo20bo29bo$21bobo10bo41b2o11bo37bo$22bo
8bo2bo16bo37bobo34bobo$50bo41b2o6b2o23b2ob2o$50b3o25b2o12b2o4bo3bo21bo
5bo$78b2o12b2o3bo5bo23bo$41bo47bobo4b2obo3bo8b2o10b2o3b2o$42bo46bo7bo
5bo8b2o$40b3o55bo3bo$100b2o$67b2o$53b2o11b4o13b2o$51b2ob2o10b2ob2o10b
2ob2o$51b4o13b2o11b4o44b2o$52b2o28b2o45bo$34bo9b2o84b3o$33bobo7b4o85bo
$26b2o4bob2o5b3o2bo2bobo39bo20b2o$26b2o3b2ob2o9b2o2bo2bo37b2o20bo$32bo
b2o6bo9b2o6b2o28bobo8bobo6bobo$33bobo5bo8bo3b2o4b2o37bo3bo6b2o$34bo6bo
10b2o33b2o10bo$49bo2bo34bo2bo7bo4bo$49bobo39bo7bo$62bo28bo7bo3bo$62b3o
26bo9bobo$65bo14b2o5bo2bo$64b2o13bobo5b2o$79bo$78b2o6$66b3o$65bo3bo$
64bo5bo8bo$64bo5bo6bobo$67bo10b2o$65bo3bo$66b3o$67bo3$67b2o$67b2o$94bo
$93bobo$91b2o3bo9b2o$91b2o3bo9b2o$91b2o3bo$87b2o4bobo$86bobo5bo$86bo$
85b2o!
p(30 × (2²² − 1)) = 125,829,090 :
Code: Select all
x = 153, y = 126, rule = B3/S23
18$93bo$92bobo$92b2obo7b2o$80b2o10b2ob2o6bobo$80b2o10b2obob3o6bo$92bob
o2bo2bo2bo2bo7b2o$93bo4b2o6bo7b2o$55bo47bobo$54bobo46b2o$53bo3b2o6b2o$
37b2o14bo3bob2o4bobo22bobo$37bobo13bo3bob3o4b3o21b2o$24bobo13bo13bobob
2o2bo4b3o6b2o13bo$23bo2bo2b2o6bo2bo14bo4b2o4b3o7b2o$22b2o5bobo8bo24bob
o$14b2o4b2o3bo3bo3bo3bobo25b2o$14b2o6b2o5b3ob2o2b2o$23bo2bo3b2o20bo24b
obo5bo$24bobo25bobo20bo3bo6bo$52b2o21bo10bo8bo$68b2o4bo4bo14b4o$40bo
27b2o5bo5bobo9b2obobo$38bobo34bo3bobobo8b3obo2bo2b2o$39b2o36bobobobo9b
2obobo3b2o$82bo11b4o$95bo35b2o$49b2o80b2o$48b4o13b2o$48b2ob2o10b2ob2o
9bo$50b2o11b4o10b2o$64b2o$37b2o39b2o$36b3o3b2obo$26b2o5bob2o5bo3bo2bo
52b2o$26b2o5bo2bo4bo4bo2b2o51bo$33bob2o4b4o5b2o8b2o30bo7bobo29bo$36b3o
3bo7b3o7b2o23b2obo3b4o4b2o29b3o$37b2o11b2o27b2o2b2obobo4b4o33b5o$49b2o
27bobo2b2obob2o3bo2bo32b2o3b2o$49bo27b3o8b2o3b4o33b5o$76b3o8b2o3b4o34b
o3bo$77b3o12bo38bobo$70b2o6bobo51bo$69bobo7b2o$69bo64b2o$68b2o64bo$
135b3o$137bo7$38bo$38b3o$41bo$40bo$40bo2bo$44bo$42bobo$43bo3$40b2o3b2o
$40bobobobo$41b5o$42b3o$43bo7$43b2o52b2o$43b2o53b2o$97bo15$89bo$88b2o$
87b2o4b2o2b2o$86b3o4b2o2b2o$87b2o4b2o$78b2o8b2o$77bobo9bo$77bo$76b2o!
p(30 × (2¹⁵³ − 1)) = 342,539,446,249,430,371,453,988,632,667,878,832,731,859,189,730:
Code: Select all
x = 787, y = 665, rule = B3/S23
34$411b2o$411bo2bo$403b5o7bo9bobo$402bo5bo6bo7bo3bo$402bo3b2o7bo7bo$
403bo7bo2bo7bo4bo8b2o$411b2o10bo12b2o$423bo3bo$414bobo8bobo$414b2o$
415bo3$370bo$370bobo$371bobo7bo26bo$352b2o17bo2bo6b2o23b2o$354bo16bobo
2b2o4b2o23b2o$341b2o12bo14bobo3b2o4b3o7b2o$341b2o4bo7bo14bo5b2o4b2o8b
2o$338b2o5b2o8bo25b2o$330b2o5b3o5bo2b2o4bo26bo$330b2o6b2o6b5ob2o15bo
31b2o$341b2o4bo20bo25bobo6bo$341b2o25b3o22bo2bo4bo$392b2o7b3o7bo$355bo
28b2o4b2o3bo4b2o8bobo$356bo27b2o6b2o5bob2o7b2obo$354b3o36bo2bob2ob2o7b
2ob2o3b2o$394bobo2bobobo6b2obo4b2o$399b4o7bobo$400b2o9bo$379b2o$366bo
11b4o14bo$363b3ob2o9b2ob2o10bo2bo$364b3o13b2o11bo$364b2o27bo2bo$353b2o
40bo$351bo2bo39bo$342b2o6bo7b3o3bo53b2o$342b2o6bo6bo3b5o52bo$350bo7bo
3b2ob2o9b2o29b2o7bobo$351bo2bo3bo3b2ob3o8b2o22bob2o3b3o6b2o$353b2o5b4o
b2o29bo2bo3bo5b2obo$362b4o29b2o2bo4bo4bo2bo$364bo29b2o5b4o4b2obo$393b
3o7bo3b3o$394b2o11b2o$386b2o7b2o$385bobo8bo$385bo$384b2o40$332bobo$
332b2o$333bo182$706b2o$706b2o6$706b2o$706b2o3$707bo$706b3o$705bo3bo$
707bo$704bo5bo$704bo5bo$122bo582bo3bo$122b3o581b3o$125bo$124bo$124bo2b
o$128bo$126bobo$127bo2$709b2o$124b2o3b2o578bo$124bobobobo579b3o$125b5o
582bo$126b3o$127bo7$127b2o$127b2o271$428bo$427b2o$426b2o4b2o2b2o$425b
3o4b2o2b2o$426b2o4b2o$417b2o8b2o$416bobo9bo$416bo$415b2o!
p(30 × (2⁴⁷¹ − 1)) = 182,914,954,120,077,669,807,515,462,683,193,338,227,456,922,421,775,653,370,653,728,103,049,125,321,470,641,803,924,185,726,376,482,685,146,109,245,394,919,310,280,154,824,958,297,967,165,410:
Code: Select all
x = 2317, y = 2053, rule = B3/S23
38$1141b2o$1140bobo$1139bo13bobo$1130b2o7bo2bo6b2o2bo2bo$1130b2o7bo8bo
bo5b2o$1140bobo3bo3bo3bo3b2o4b2o$1141b2o2b2ob3o5b2o6b2o$1148b2o3bo2bo$
1153bobo3$1139bo$1139bobo$1095bobo41b2o$1093bo3bo$1093bo19bo$1085bo6bo
4bo14b4o$1084bobo6bo12bo4b2obobo$1067b2o14bo3b2o4bo3bo3bob2o5b3obo2bo
2b2o10bo$1067bobo13bo3b2o6bobo2bob4o5b2obobo3b2o9bo$1062b2o4b3o12bo3b
2o11b2o2b2o6b4o15b3o$1058b4o2bo4b3o12bobo26bo$1058b3ob2o4b3o7bo6bo$
1067bobo9b2o40b2o$1067b2o9b2o34b3o4b2o$1113b5o6b2o10bo$1093bobo16bobo
3bo5b3o7bobo$1093b2o17b2o3bo6b2o6b2o$1094bo26b2o9b2o12b2o$1121b2o9b2o
12b2o$1085bobo46bobo$1086b2o48bo$1086bo25bo2bo$1098b4o14bo$1097bo3bo
10bo3bo$1101bo11b4o$1097bo2bo$1079b2o9b2o$1079bobo7bobo$1070b2o2b2o6bo
8bo5bo48b2o$1070b2obo2bo2bo2bo13bobo47bo$1074b2o6bo13b2obo4b2o19b3o9b
2o5bobo$1079bobo14b2ob2o3b2o19bo9bo2bo5b2o$1079b2o15b2obo20bo5bo7bo$
1096bobo19b4o12bo$1097bo19bobob2o11bo$1116bo2bob3o11bo2bo$1116b2obob2o
14b2o$1114b3ob4o$1113bobo4bo$1113bo$1112b2o178$922bo$922bobo$922b2o
652$2029b2o$2029b2o3$255bo$255b3o$258bo$257b2o4$2029b3o$2028bo3bo$
2027bo5bo$257b2o3b2o1764bo3bo$257b2o3b2o1765b3o$258b5o1766b3o$259bobo
2$259b3o1770b2o$2032bo$2033b3o$2035bo3$260b2o$260b2o869$1155bo$1154bob
o$1153bob2o10b2o$1152b2ob2o10b2o$1153bob2o$1148b2o4bobo$1147bobo5bo$
1147bo$1146b2o!
Reminder:
- n is the number of bits in the loop.
- S = {n ≥ 2 | Pₙ(x) = (x + 1)ⁿ + x⁽ⁿ⁻¹⁾ is irreducible and primitive over F₂}
- For each n belonging to S: P(n) = 30 × (2ⁿ − 1)