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 NPb to avoid clashing with existing letter conventions for groups.
Prefer G or S? Replace them in your local copy.
Conjecture: Order of 2 mod N when N of the form 1+(1+2^t)*(2^t) is 3×t
When t=3 we have P=1+9×8 and the set of elements modulo P is a multiplicative group.
The subgroup generated by 2 has 9 elements
The elements in NPb9 are as shown next
{ 2, 4, 8, 16, 32, 64, 55, 37, 1 }
If NPb9 is really a group then we need inverses so let us document those next.
- Inverse of 2 is 37 mod P
- Inverse of 4 is 55 mod P
- Inverse of 8 is 64 mod P
- Inverse of 16 is 32 mod P
From 273 for t=4 the 12 elements in NPb12 are as shown next
{ 2, 4, 8, 16, 32, 64, 128, 256, 239, 205, 137, 1 }
From 1057 for t=5 the 15 elements in NPb15 are as shown next
{ 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024,
991, 925, 793, 529, 1 }
From 4161 for t=6 the 18 elements in NPb18 are as shown next
{ 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024,
2048, 4096, 4031, 3901, 3641, 3121, 2081, 1 }
From 16513 for t=7 the 21 elements in NPb21 are as shown next
{ 2, 4, 8, 16, 32, 64, 128,
256, 512, 1024, 2048, 4096, 8192, 16384,
16255, 15997, 15481, 14449, 12385, 8257, 1 }
From 65793 for t=8 the 24 elements in NPb24 are as shown next
{ 2, 4, 8, 16, 32, 64, 128, 256,
512, 1024, 2048, 4096, 8192, 16384, 32768, 65536,
65279, 64765, 63737, 61681, 57569, 49345, 32897, 1 }
For t=9 the 27 elements in NPb27 are as shown next
{ 2, 4, 8, 16, 32, 64, 128, 256, 512,
1024, 2048, 4096, 8192, 16384, 32768, 65536, 131072, 262144,
261631, 260605, 258553, 254449, 264241, 229825, 196993,
131329, 1 }
For t=10 the 30 elements in NPb30 are as shown next
{ 2, 4, 8, 16, 32, 64, 128,
256, 512, 1024, 2048, 4096, 8192,
16384, 32768, 65536, 131072, 262144,
524288, 1048576, 1047551, 1045501, 1041401, 1033201,
1016801, 984001, 918401, 787201, 524801, 1 }
For t=11 the 33 elements in NPb33 are not tabulated here
For t=12 the 36 elements in NPb36 are as shown next
{ 2, 4, 8, 16, 32, 64, 128,
256, 512, 1024, 2048, 4096, 8192,
16384, 32768, 65536, 131072, 262144,
524288, 1048576, 2097152, 4194304, 8388608, 16777216,
16773119, 16764925, 16748537, 16715761, 16650209, 16519105,
16256897, 15732481, 14683649, 12585985, 8390657, 1 }
For t=13 the 39 elements in NPb39 are not tabulated here
For t=14 the 42 elements in NPb42 are not tabulated here
For t=15 the 45 elements in NPb45 are not tabulated here
For t=16 the 48 elements in NPb48 are not tabulated here
For t=17 the 51 elements in NPb51 are not tabulated here
For t=18 the 54 elements in NPb54 are not tabulated here
For t=19 the 57 elements in NPb57 are not tabulated here
For t=20 the 60 elements in NPb60 are as shown next
{ 2, 4, 8, 16, 32,
64, 128, 256, 512, 1024,
2048, 4096, 8192, 16384, 32768,
65536, 131072, 262144, 524288, 1048576,
2097152, 4194304, 8388608, 16777216, 33554432,
67108864, 134217728, 268435456, 536870912, 1073741824,
2147483648, 4294967296, 8589934592,
17179869184, 34359738368, 68719476736,
137438953472, 274877906944, 549755813888,
1099511627776, 1099510579199, 1099508482045,
1099504287737, 1099495899121, 1099479121889,
1099445567425, 1099378458497, 1099244240641, 1098975804929, 1098438933505, 1097365190657,
1095217704961, 1090922733569, 1082332790785,
1065152905217, 1030793134081, 962073591809,
824634507265, 549756338177, 1 }
Using sturdy element notation we might say the results tabulated in this post are
