Consider the space
We will be using generating element 2
The groups generated by 2 have low order [ a lot lower than n-1 ]
Tabulating the elements of groups generated by 2 next
Using prefix NNa to avoid clashing with existing letter conventions for groups.
Prefer G or S? Replace them in your local copy.
From t=3 the elements in NNa18 are
{ 2, 4, 8, 16, 32, 7, 14, 28, -1,
55, 53, 49, 41, 25, 50, 43, 29, 1 }
Conjecture: Order of 2 mod N when N of the form 1+(-1+2^t)*(2^t) is 6×t
When t=4 we have P=1+15×16 and the set of elements modulo P is a multiplicative group.
The subgroup generated by 2 has 24 elements
The elements in NNa24 are as shown next
{ 2, 4, 8, 16, 32, 64, 128, 15, 30, 60, 120, -1
239, 237, 233, 225, 209, 177, 113, 226, 211, 181, 121, 1 }
If NNa24 is really a group then we need inverses so let us document those next.
- Inverse of 2 is 121 mod P
- Inverse of 4 is 181 mod P
- Inverse of 8 is 211 mod P
- Inverse of 16 is 226 mod P
- Inverse of 32 is 113 mod P
- Inverse of 64 is 177 mod P
- Inverse of 128 is 209 mod P
- Inverse of 15 is 225 mod P
- Inverse of 30 is 233 mod P
- Inverse of 60 is 237 mod P
- Inverse of 120 is 239 mod P
We could alternatively write the group NNa24 as follows:
{ 2, 4, 8, 16, 32, 64, (2^7), (2^8), (2^9), (2^10), (2^11), -1,
-2, -4, -8, -16, -32, -64, -(2^7), -(2^8), -(2^9), -(2^10), -(2^11), 1 }
From t=5 the 30 elements in NNa30 are as shown next
{ 2, 4, 8, 16, 32, 64, 128, 256, 512, 31,
62, 124, 248, 496, -1, 991, 989, 985, 977, 961,
929, 865, 737, 481, 962, 931, 869, 745, 497, 1 }
From t=6 the 36 elements in NNa36 are as shown next
{ 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024,
2048, 63, 126, 252, 504, 1008, 2016, -1, 4031, 4029,
4025, 4017, 4001, 3969, 3905, 3777, 3521, 3009, 1985, 3970,
2907, 3781, 3529, 3025, 2017, 1 }
From t=7 the 42 elements in NNa42 are as shown next.
{ 2, 4, 8, 16, 32, 64, 128,
256, 512, 1024, 2048, 4096, 8192, 127,
254, 508, 1016, 2032, 4064, 8128, -1,
16255, 16253, 16249, 16241, 16225, 16193, 16129,
16001, 15745, 15233, 14209, 12161, 8065, 16130,
16003, 15749, 15241, 14225, 12193, 8129, 1 }
From t=8 the 48 elements in NNa48 are as shown next.
{ 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024,
2048, 4096, 8192, 16384, 32768, 255, 510, 1020, 2040, 4080,
8160, 16320, 32640, -1, 65279, 65277, 65273, 65265, 65249, 65217,
65153, 65025, 64769, 64257, 63233, 57089, 48897, 32513, 65026,
64771, 64261, 63241, 61201, 57121, 48961, 32641, 1 }
From t=9 the 54 elements in NNa54 are as shown next.
{ 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024,
2048, 4096, 8192, 16384, 32768, 65536, 131072, 511, 1022, 2044,
4088, 8176, 16352, 32704, 65408, 130816, -1, 261631,
261629, 261625, 261617, 261601, 261569, 261505, 261377, 261121,
260609, 259585, 257537, 253441, 245249, 228865,
196097, 130561, 261122, 260611, 259589, 257545,
253457, 245281, 228929, 196225, 130817, 1 }
From t=10 the 60 elements in NNa60 are as shown next.
{ 2, 4, 8, 16, 32, 64, 128, 256,
512, 1024, 2048, 4096, 8192, 16384, 32768, 65536,
131072, 262144, 524288, 1023, 2046, 4092, 8184, 16368,
32736, 65472, 130944, 261888, 523776, -1, 1047551, 1047549,
1047545, 1047537, 1047521, 1047489, 1047425, 1047297, 1047041, 1046529, 1045505, 1043457, 1039361, 1031169, 1014785,
982017, 916481, 785409, 523265, 1046530, 1045507,
1043461, 1039369, 1031185, 1014817, 982081,
916609, 785665, 523777, 1 }
For t=11 the 66 elements in NNa66 are not tabulated here
For t=12 the 72 elements in NNa72 are not tabulated here
For t=13 the 78 elements in NNa78 are not tabulated here
For t=14 the 84 elements in NNa84 are not tabulated here
For t=15 the 90 elements in NNa90 are not tabulated here
For t=16 the 96 elements in NNa96 are not tabulated here
For t=17 the 102 elements in NNa102 are not tabulated here
For t=18 the 108 elements in NNa108 are not tabulated here
For t=19 the 114 elements in NNa114 are not tabulated here
For t=20 the 120 elements in NNa120 are not tabulated here
Using sturdy element notation we might say the results tabulated in this post are
