Consider the Proth space
We will be using generating element 3
The [sub]groups generated by 3 have low order [ a lot lower than n-1 ]
Tabulating the elements of groups generated by 3 next
Using prefix ENp 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+1)) is 6×(1+2×t)
When t=2 we have P=1+8×27 and the set of elements modulo P is a multiplicative group.
The subgroup generated by 3 has 30 elements
The elements in ENp30 are as shown next
{ 3, 9, 27, 81, 26, 78, 17, 51, 153, 25,
75, 8, 24, 72, -1, 214, 208, 190, 136, 191,
139, 200, 166, 64, 192, 142, 209, 193, 145, 1 }
If ENp30 is really a group then we need inverses so let us document those next.
- Inverse of 3 is 145 mod P
- Inverse of 9 is 193 mod P
- Inverse of 27 is 209 mod P
- Inverse of 81 is 142 mod P
- Inverse of 26 is 192 mod P
- Inverse of 78 is 64 mod P
- Inverse of 17 is 166 mod P
- Inverse of 51 is 200 mod P
- Inverse of 153 is 139 mod P
- Inverse of 25 is 191 mod P
- Inverse of 75 is 136 mod P
- Inverse of 8 is 190 mod P
- Inverse of 24 is 208 mod P
- Inverse of 72 is 214 mod P \\ 214==Mod((3^14),pr)^-1
We could alternatively write the group ENp30 as follows:
{ 3, 9, 27, 81, (3^5), (3^6), (3^7), (3^8), (3^9), (3^10),
(3^11), -(3^12), -(3^13), -(3^14), -1,
-3, -9, -27, -81, -(3^5), -(3^6), -(3^7), -(3^8), -(3^9), -(3^10),
-(3^11), -(3^12), -(3^13), -(3^14), 1 }
From t=3 the 42 elements in ENp42 are as shown next
{ 3, 9, 27, 81, 243, 729, 80, 240, 720, 53
159, 477, 1431, 79, 237, 711, 26, 78, 234, 702,
-1, 2104, 2098, 2080, 2026, 1864, 1378, 2027, 1867, 1387,
2054, 1948, 1630, 676, 2028, 1870, 1396, 2081, 2029, 1873,
1405, 1 }
From t=4 the 54 elements in ENp54 are as shown next
{ 3, 9, 27, 81, 243, 729, 2187, 6561,
242, 726, 2178, 6534, 161, 483, 1449, 4347,
13041, 241, 723, 2169, 6507, 80, 240, 720,
2160, 6480, -1, 19438, 19432, 19414, 19360, 19198,
18712, 17254, 12880, 19199, 18715, 17263, 12907, 19280,
18958, 17992, 15094, 6400, 19200, 18718, 17272, 12934,
19361, 19201, 18721, 17281, 12961, 1 }
From t=5 the 66 elements in ENp66 are as shown next.
{ 3, 9, 27, 81, 243, 729, 2187, 6561,
19683, 59049, 728, 2184, 6552, 19656, 58968, 485,
1455, 4365, 13095, 39285, 117855, 727, 2181, 6543,
19629, 58887, 242, 726, 2178, 6534, 19602, 58806,
-1, 176416, 176410, 176392, 176338, 176176, 175690, 174232,
169858, 156736, 117370, 175691, 174235, 169867, 156763, 117451,
175934, 174964, 172054, 163324, 137134, 58564, 175692, 174238,
169876, 156790, 117532, 176177,
175693, 174241, 169885, 156817, 117613, 1 }
From t=6 the 78 elements in ENp78 are not tabulated here
From t=7 the 90 elements in ENp90 are not tabulated here
From t=8 the 102 elements in ENp102 are not tabulated here
From t=9 the 114 elements in ENp114 are not tabulated here
For t=10 the 126 elements in ENp126 are not tabulated here
For t=11 the 138 elements in ENp138 are not tabulated here
For t=12 the 150 elements in ENp150 are not tabulated here
For t=13 the 162 elements in ENp162 are not tabulated here
For t=14 the 174 elements in ENp174 are not tabulated here
For t=15 the 186 elements in ENp186 are not tabulated here
For t=16 the 198 elements in ENp198 are not tabulated here
For t=17 the 210 elements in ENp210 are not tabulated here
For t=18 the 222 elements in ENp222 are not tabulated here
For t=19 the 234 elements in ENp234 are not tabulated here
For t=20 the 246 elements in ENp246 are not tabulated here