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 NNe 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 6×t
When t=2 we have P=1+8×9 and the set of elements modulo P is a multiplicative group.
The subgroup generated by 3 has 12 elements
The elements in NNe12 are { 3, 9, 27, 8, 24, -1, 70, 64, 46, 65, 49, 1 }
If NNe12 is really a group then we need inverses so let us document those next.
- Inverse of 3 is 49 mod P
- Inverse of 9 is 65 mod P
- Inverse of 27 is 46 mod P
- Inverse of 8 is 64 mod P
- Inverse of 24 is 70 mod P
We could alternatively write the group NNe12 as follows:
{ 3, 9, 27, (3^4), (3^5), -1,
-3, -9, -27, -(3^4), -(3^5), 1, }
For 703 from t=3 the 18 elements in NNe18 are as shown next
{ 3, 9, 27, 81, 243, 26, 78, 234, -1
700, 694, 676, 622, 460, 677, 625, 469, 1 }
For 6481 from t=4 the 24 elements in NNe24 are as shown next
{ 3, 9, 27, 81, 243, 729, 2187, 80, 240, 720,
2160, -1, 6478, 6472, 6454, 6400, 6238, 5752, 4294, 6401,
6241, 5761, 4321, 1 }
For 58807 from t=5 the 30 elements in NNe30 are as shown next
{ 3, 9, 27, 81, 243, 729,
2187, 6561, 19683, 242, 726, 2178,
6534, 19602, -1, 58804, 58798, 58780,
58726, 58564, 58078, 56620, 52246, 39124,
58565, 58081, 56629, 52273, 39205, 1 }
For 530713 from t=6 the 36 elements in NNe36 are as shown next
{ 3, 9, 27, 81, 243, 729,
2187, 6561, 19683, 59049, 177147, 728,
2184, 6552, 19656, 58968, 176904, -1,
530710, 530704, 530686, 530632, 530470, 529984,
528526, 524152, 511030, 471664, 353566, 529985,
528529, 524161, 511057, 471745, 353809, 1 }
For 4780783 from t=7 the 42 elements in NNe42 are as shown next
{ 3, 9, 27, 81, 243, 729,
2187, 6561, 19683, 59049, 177147, 531441,
1594323, 2186, 6558, 19674, 59022, 177066,
531198, 1593594, -1, 4780780, 4780774, 4780756,
4780702, 4780540, 4780054, 4778596, 4774222, 4761100,
4721734, 4603636, 4249342, 3186460, 4778597, 4774225,
4761109, 4721761, 4603717, 4249585, 3187189, 1 }
For t=8 the 48 elements in NNe48 are not tabulated here
For t=9 the 54 elements in NNe54 are not tabulated here
For t=10 the 60 elements in NNe60 are not tabulated here
For t=11 the 66 elements in NNe66 are not tabulated here
For t=12 the 72 elements in NNe72 are not tabulated here
For t=13 the 78 elements in NNe78 are not tabulated here
For t=14 the 84 elements in NNe84 are not tabulated here
For t=15 the 90 elements in NNe90 are not tabulated here
For t=16 the 96 elements in NNe96 are not tabulated here
For t=17 the 102 elements in NNe102 are not tabulated here
For t=18 the 108 elements in NNe108 are not tabulated here
For t=19 the 114 elements in NNe114 are not tabulated here
For t=20 the 120 elements in NNe120 are not tabulated here
Using sturdy element notation we might say the results tabulated in this post are
