Consider the space
We will be using generating element 5
The groups generated by 5 have low order [ a lot lower than n-1 ]
Tabulating the elements of groups generated by 5 next
Using prefix NNd to avoid clashing with existing letter conventions for groups.
Prefer G or S? Replace them in your local copy.
Conjecture: Order of 5 mod N when N of the form 1+(-1+5^t)*(5^t) is 6×t
When t=2 we have P=1+24×25 and the set of elements modulo P is a multiplicative group.
The multiplicative group generated by 5 has 12 elements
The elements in NNd12 are
{ 5, 25, 125, 24, 120, -1, 596, 576, 476, 577, 481, 1 }
If NNd12 is really a group then we need inverses so let us document those next.
- Inverse of 5 is 481 mod P
- Inverse of 25 is 577 mod P
- Inverse of 125 is 476 mod P
- Inverse of 24 is 576 mod P
- Inverse of 120 is 596 mod P
We could alternatively write the group NNd12 as follows:
{ 5, 25, 125, (5^4), (5^5), -1,
-5, -25, -125, -(5^4), -(5^5), 1, }
For 15501 from t=3 the 18 elements in NNd18 are as shown next
{ 5, 25, 125, 625, 3125, 124, 620, 3100, -1,
15496, 15476, 15376, 14876, 12376, 15377, 14881, 12401, 1 }
For 390001 from t=4 the 24 elements in NNd24 are as shown next
{ 5, 25, 125, 625, 3125, 15625, 78125, 624,
3120, 15600, 78000, -1, 389996, 389976, 389876, 389376,
386876, 374376, 311876, 389377, 386881, 374401, 312001, 1 }
For 9762501 from t=5 the 30 elements in NNd30 are as shown next
{ 5, 25, 125, 625, 3125, 15625,
78125, 390625, 1953125, 3124, 15620, 78100,
390500, 1952500, -1, 9762496, 9762476, 9762376,
9761876, 9759376, 9746876, 9684376, 9371876, 7809376,
9759377, 9746881, 9684401, 9372001, 7810001, 1 }
For 244125001 from t=6 the 36 elements in NNd36 are as shown next
{ 5, 25, 125, 625, 3125, 15625,
78125, 390625, 1953125, 9765625, 48828125, 15624,
78120, 390600, 1953000, 9765000, 48825000, -1,
244124996, 244124976, 244124876,
244124376, 244121876, 244109376,
244046876, 243734376, 242171876,
234359376, 195296876, 244109377,
244046881, 243734401, 242172001,
234360001, 195300001, 1 }
For 6103437501 from t=7 the 42 elements in NNd42 are as shown next
{ 5, 25, 125, 625, 3125, 15625,
78125, 390625, 1953125, 9765625, 48828125, 244140625,
1220703125, 78124, 390620, 1953100, 9765500, 48827500,
244137500, 1220687500, -1,
6103437496, 6103437476, 6103437376,
6103436876, 6103434376, 6103421876,
6103359376, 6103046876, 6101484376,
6093671876, 6054609376, 5859296876,
4882734376, 6103359377, 6103046881,
6101484401, 6093672001, 6054610001,
5859300001, 4882750001, 1 }
For t=8 the 48 elements in NNd48 are not tabulated here
For t=9 the 54 elements in NNd54 are not tabulated here
For t=10 the 60 elements in NNd60 are not tabulated here
For t=11 the 66 elements in NNd66 are not tabulated here
For t=12 the 72 elements in NNd72 are not tabulated here
For t=13 the 78 elements in NNd78 are not tabulated here
For t=14 the 84 elements in NNd84 are not tabulated here
For t=15 the 90 elements in NNd90 are not tabulated here
For t=16 the 96 elements in NNd96 are not tabulated here
For t=17 the 102 elements in NNd102 are not tabulated here
For t=18 the 108 elements in NNd108 are not tabulated here
For t=19 the 114 elements in NNd114 are not tabulated here
For t=20 the 120 elements in NNd120 are not tabulated here
Using sturdy element notation we might say the results tabulated in this post are
