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 TNx to avoid clashing with existing letter conventions for groups.
Prefer G or S? Replace them in your local copy.
From t=2 the elements in TNx20 are as shown next
{ 2, 4, 8, 16, 7, 14, 3, 6, 12, -1,
23, 21, 17, 9, 18, 11, 22, 19, 13, 1 }
Conjecture: Order of 2 mod N when N of the form 1+(-1+2^t)*(2^(t+1)) is 4×(1+2×t)
When t=3 we have P=1+7×16 and the set of elements modulo P is a multiplicative group.
The subgroup generated by 2 has 28 elements
The elements in TNx28 are as shown next
{ 2, 4, 8, 16, 32, 64, 15, 30, 60, 7, 14, 28, 56, -1,
111, 109, 105, 97, 81, 49, 98, 83, 53, 106, 99, 85, 57, 1 }
If TNx28 is really a group then we need inverses so let us document those next.
- Inverse of 2 is 57 mod P
- Inverse of 4 is 85 mod P
- Inverse of 8 is 99 mod P
- Inverse of 16 is 106 mod P
- Inverse of 32 is 53 mod P
- Inverse of 15 is 98 mod P
- Inverse of 30 is 49 mod P
- Inverse of 60 is 81 mod P
- Inverse of 7 is 97 mod P
- Inverse of 14 is 105 mod P
- Inverse of 28 is 109 mod P
- Inverse of 56 is 111 mod P
We could alternatively write the group TNx28 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 TNx36 are as shown next
{ 2, 4, 8, 16, 32, 64, 128, 256, 31,
62, 124, 248, 15, 30, 60, 120, 240, -1,
479, 477, 473, 465, 449, 417, 353, 225, 450,
419, 357, 233, 466, 451, 421, 361, 241, 1 }
From t=5 the 44 elements in TNx44 are as shown next
{ 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024,
63, 126, 252, 504, 1008, 31, 62, 124, 248, 496,
992, -1, 1983, 1981, 1977, 1969, 1953, 1921, 1857, 1729
1473, 961, 1922, 1859, 1733, 1481, 977, 1954, 1923, 1861,
1737, 1489, 993, 1 }
From t=6 the 52 elements in TNx52 are as shown next.
{ 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024,
2048, 4096, 127, 254, 508, 1016, 2032, 4064, 63, 126,
252, 504, 1008, 2016, 4032, -1, 8063, 8061, 8057, 8049,
8033, 8001, 7937, 7809, 7553, 7041, 6017, 3969, 7938, 7811,
7557, 7049, 6033, 4011, 8002, 7939, 7813, 7561, 7057, 6049,
4033, 1 }
From t=7 the 60 elements in TNx60 are as shown next.
{ 2, 4, 8, 16, 32, 64,
128, 256, 512, 1024, 2048, 4096,
8192, 16384, 255, 510, 1020, 2040,
4080, 8160, 16320, 127, 254, 508,
1016, 2032, 4064, 8128, 16256, -1,
32511, 32509, 32505, 32497, 32481, 32449,
32385, 32257, 32001, 31489, 30465, 28417,
24321, 16129, 32258, 32003, 31493, 30473,
28433, 24353, 16193, 32386, 32259, 32005,
31497, 30481. 28449, 24385, 16257, 1 }
From t=8 the 68 elements in TNx68 are as shown next.
{ 2, 4, 8, 16, 32, 64,
128, 256, 512, 1024, 2048, 4096,
8192, 16384, 32768, 65536, 511, 1022,
2044, 4088, 8176, 16352, 32704, 65408,
255, 510, 1020, 2040, 4080, 8160,
16320, 32640, 65280, -1, 130559, 130557,
130553, 130545, 130529, 130497, 130433, 130305,
130049, 129537, 128513, 126465, 122369, 114177,
97793, 65025, 130050, 129539, 128517, 126473,
122385, 114209, 97857, 65153, 130306, 130051,
129541, 128521, 126481, 122401, 114241, 97921, 65281, 1 }
From t=9 the 76 elements in TNx76 are not tabulated here
For t=10 the 84 elements in TNx84 are not tabulated here
For t=11 the 92 elements in TNx92 are not tabulated here
For t=12 the 100 elements in TNx100 are not tabulated here
For t=13 the 108 elements in TNx108 are not tabulated here
For t=14 the 116 elements in TNx116 are not tabulated here
For t=15 the 124 elements in TNx124 are not tabulated here
For t=16 the 132 elements in TNx132 are not tabulated here
For t=17 the 140 elements in TNx140 are not tabulated here
For t=18 the 148 elements in TNx148 are not tabulated here
For t=19 the 156 elements in TNx156 are not tabulated here
For t=20 the 164 elements in TNx164 are not tabulated here
Using sturdy element notation we might say the results tabulated in this post are
