Consider the space
We will be using generating element 3
The groups generated by 3 have low order [ a lot lower than n-1 ]
Tabulating the elements of groups generated by 3 next
Using prefix NPf to avoid clashing with existing letter conventions for groups.
Prefer G or S? Replace them in your local copy.
Conjecture: Order of 3 mod N when N of the form 1+(1+3^t)*(3^t) is 3×t
When t=3 we have P=1+28×27 and the set of elements modulo P is a multiplicative group.
The subgroup generated by 3 has 9 elements
The elements in NPf9 are as shown next
{ 3, 9, 27, 81, 243, 729, 673, 505, 1 }
If NPf9 is really a group then we need inverses so let us document those next.
- Inverse of 3 is 505 mod P
- Inverse of 9 is 673 mod P
- Inverse of 27 is 729 mod P
- Inverse of 81 is 243 mod P
For 6643 from t=4 the 12 elements in NPf12 are as shown next
{ 3, 9, 27, 81, 243, 729,
2187, 6561, 6397, 5905, 4429, 1 }
For 59293 from t=5 the 15 elements in NPf15 are as shown next
{ 3, 9, 27, 81, 243,
729, 2187, 6561, 19683, 59049,
58561, 57097, 52705, 39529, 1 }
For 532171 from t=6 the 18 elements in NPf18 are as shown next
{ 3, 9, 27, 81, 243, 729,
2187, 6561, 19683, 59049, 177147, 531441,
529981, 525601, 512461, 473041, 354781, 1 }
For 4785157 from t=7 the 21 elements in NPf21 are as shown next
{ 3, 9, 27, 81, 243, 729, 2187,
6561, 19683, 59049, 177147, 531441, 1594323, 4782969,
4778593, 4765465, 4726081, 4607929, 4253473, 3190105, 1 }
For 43053283 from t=8 the 24 elements in NPf24 are as shown next
{ 3, 9, 27, 81, 243, 729,
2187, 6561, 19683, 59049, 177147, 531441,
1594323, 4782969, 14348907, 43046721, 43033597, 42994225,
42876109, 42521761, 41458717, 38269585, 28702189, 1 }
For 387440173 from t=9 the 27 elements in NPf27 are as shown next
{ 3, 9, 27, 81, 243, 729,
2187, 6561, 19683, 59049, 177147, 531441,
1594323, 4782969, 14348907, 43046721, 129140163,
387420489, 387381121, 387263017, 386908705, 385845769,
382656961, 373090537, 344391265, 258293449, 1 }
For 3486843451 from t=10 the 30 elements in NPf30 are as shown next
{ 3, 9, 27, 81, 243,
729, 2187, 6561, 19683, 59049,
177147, 531441, 1594323, 4782969, 14348907,
43046721, 129140163, 387420489, 1162261467, 3486784401,
3486666301, 3486312001, 3485249101, 3482060401, 3472494301,
3443796001, 3357701101, 3099416401, 2324562301, 1 }
For 31381236757 from t=11 the 33 elements in NPf33 are as shown next
{ 3, 9, 27, 81, 243,
729, 2187, 6561, 19683, 59049,
177147, 531441, 1594323, 4782969, 14348907,
43046721, 129140163, 387420489, 1162261467, 3486784401,
10460353203, 31381059609, 31380705313, 31379642425, 31376453761, 31366887769, 31338189793, 31252095865,
30993814081, 30218968729, 27894432673, 20920824505, 1 }
For 282430067923 from t=12 the 36 elements in NPf36 are as shown next
{ 3, 9, 27, 81, 243, 729, 2187, 6561, 19683, 59049,
177147, 531441, 1594323, 4782969, 14348907,
43046721, 129140163, 387420489, 1162261467, 3486784401,
10460353203, 31381059609, 94143178827, 282429536481,
282428473597, 282425284945, 282415718989, 282387021121,
282300927517, 282042646705, 281267804269, 278943276961,
271969695037, 251048949265, 188286711949, 1 }
For t=13 the 39 elements in NPf39 are not tabulated here
For t=14 the 42 elements in NPf42 are not tabulated here
For t=15 the 45 elements in NPf45 are not tabulated here
For t=16 the 48 elements in NPf48 are not tabulated here
For t=17 the 51 elements in NPf51 are not tabulated here
For t=18 the 54 elements in NPf54 are not tabulated here
For t=19 the 57 elements in NPf57 are not tabulated here
For t=20 the 60 elements in NPf60 are not tabulated here
Using sturdy element notation we might say the results tabulated in this post are
