Consider the Proth space
We will be using generating element 2
The [sub]groups generated by 2 have low order [ a lot lower than n-1 ]
Tabulating the elements of groups generated by 2 next.
Using the prefix TPv to avoid clashing with existing letter conventions for groups.
Prefer G or S? Replace them in your local copy.
When t=2 we have P=1+5×8 and the set of elements modulo P is a multiplicative group.
The subgroup generated by 2 has 20 elements
The elements in TPv20 are as shown next
{ 2, 4, 8, 16, 32, 23, 5, 10, 20, -1
39, 37, 33, 25, 9, 18, 36, 31, 21, 1 }
If TPv20 is really a group then we need inverses so let us document those next.
- Inverse of 2 is 21 mod P
- Inverse of 4 is 31 mod P
- Inverse of 8 is 36 mod P
- Inverse of 16 is 18 mod P
- Inverse of 32 is 9 mod P
- Inverse of 23 is 25 mod P
- Inverse of 5 is 33 mod P
- Inverse of 10 is 37 mod P
- Inverse of 20 is 39 mod P
Conjecture: Order of 2 mod N when N of the form 1+(1+2^t)*(2^(t+1)) is 4×(1+2×t)
From t=3 the 28 elements in TPv28 are as shown next
{ 2, 4, 8, 16, 32, 64, 128, 111, 77, 9,
18, 36, 72, -1, 143, 141, 137, 129, 113, 81,
17, 34, 68, 136, 127, 109, 73, 1 }
We could alternatively write the group TPv28 as follows:
{ 2, 4, 8, 16, 32, 64, 15, 30, 60, 7, 14, 28, 56, -1,
-2, -4, -8, -16, -32, -64, -(2^7), -(2^8), -(2^9), -(2^10), -(2^11),
-(2^12), -(2^13), 1 }
From t=4 the 36 elements in TPv36 are as shown next
{ 2, 4, 8, 16, 32, 64, 128, 256, 512, 479,
413, 281, 17, 34, 68, 136, 272, -1, 543, 541,
537, 529, 513, 481, 417, 289, 33, 66, 132, 264,
528, 511, 477, 409, 273, 1 }
From t=5 the 44 elements in TPv44 are as shown next
{ 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024,
2048, 1983, 1853, 1593, 1073, 33, 66, 132, 264, 528,
1056, -1, 2111, 2109, 2105, 2097, 2081, 2049, 1985, 1857,
1601, 1089, 65, 130, 260, 520, 1040, 2080, 2047, 1981,
1849, 1585, 1057, 1 }
From t=6 the 52 elements in TPv52 are as shown next.
{ 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024,
2048, 4096, 8192, 8063, 7805, 7289, 6257, 4193, 65, 130,
260, 520, 1040, 2080, 4160, -1, 8319, 8317, 8313, 8305,
8289, 8257, 8193, 8065, 7809, 7297, 6273, 4225, 129, 258,
516, 1032, 2064, 4128, 8256, 8191, 8061, 7801, 7281, 6241,
4161, 1 }
From t=7 the 60 elements in TPv60 are as shown next.
{ 2, 4, 8, 16, 32, 64,
128, 256, 512, 1024, 2048, 4096,
8192, 16384, 32768, 32511, 31997, 30969,
28913, 24801, 16577, 129, 258, 516,
1032, 2064, 4128, 8256, 16512, -1,
33023, 33021, 33017, 33009, 32993, 32961,
32897, 32769, 32513, 32001, 30977, 28929,
24833, 16641, 257, 514, 1028, 2056,
4112, 8224, 16448, 32896, 32767, 32509,
31993, 30961, 28897, 24769, 16513, 1 }
From t=8 the 68 elements in TPv68 are as shown next.
{ 2, 4, 8, 16, 32, 64,
128, 256, 512, 1024, 2048, 4096,
8192, 16384, 32768, 65536, 131072, 130559,
129533, 127481, 123377, 115169, 98753, 65921,
257, 514, 1028, 2056, 4112, 8224,
16448, 32896, 65792, -1, 131583, 131581,
131577, 131569, 131553, 131521, 131457, 131329,
131073, 130561, 129537, 127489, 123393, 115201,
98817, 66049, 513, 1026, 2052, 4104,
8208, 16416, 32832, 65664, 131328, 131071,
130557, 129529, 127473, 123361, 115137, 98689, 65793, 1 }
From t=9 the 76 elements in TPv76 are not tabulated here
For t=10 the 84 elements in TPv84 are not tabulated here
For t=11 the 92 elements in TPv92 are not tabulated here
For t=12 the 100 elements in TPv100 are not tabulated here
For t=13 the 108 elements in TPv108 are not tabulated here
For t=14 the 116 elements in TPv116 are not tabulated here
For t=15 the 124 elements in TPv124 are not tabulated here
For t=16 the 132 elements in TPv132 are not tabulated here
For t=17 the 140 elements in TPv140 are not tabulated here
For t=18 the 148 elements in TPv148 are not tabulated here
For t=19 the 156 elements in TPv156 are not tabulated here
For t=20 the 164 elements in TPv164 are not tabulated here