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 NPh 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 3×t
When t=1 we have P=1+6×5 and the set of elements modulo P is a multiplicative group.
The subgroup generated by 5 has 3 elements
The elements in NPh3 are { 5, 25, 1 }
If NPh3 is really a group then we need inverses so let us document those next.
- Inverse of 5 is 25 mod P
For 651 from t=2 the 6 elements in NPh6 are as shown next
{ 5, 25, 125, 625, 521, 1 }
For 15751 from t=3 the 9 elements in NPh9 are as shown next
{ 5, 25, 125, 625, 3125, 15625, 15121, 12601, 1 }
For 391251 from t=4 the 12 elements in NPh12 are as shown next
{ 5, 25, 125, 625, 3125, 15625,
78125, 390625, 388121, 375601, 313001, 1 }
For 9768751 from t=5 the 15 elements in NPh15 are as shown next
{ 5, 25, 125, 625, 3125, 15625,
78125, 390625, 1953125, 9765625, 9753121, 9690601,
9378001, 7815001, 1 }
For 244156251 from t=6 the 18 elements in NPh18 are as shown next
{ 5, 25, 125, 625, 3125, 15625,
78125, 390625, 1953125, 9765625, 48828125, 244140625,
244078121, 243765601, 242203001, 234390001, 195325001, 1 }
For 6103593751 from t=7 the 21 elements in NPh21 are as shown next
{ 5, 25, 125, 625, 3125, 15625,
78125, 390625, 1953125, 9765625, 48828125, 244140625,
1220703125, 6103515625, 6103203121,
6101640601, 6093828001, 6054765001,
5859450001, 4882875001, 1 }
For 152588281251 from t=8 the 24 elements in NPh24 are as shown next
{ 5, 25, 125, 625, 3125,
15625, 78125, 390625, 1953125, 9765625,
48828125, 244140625, 1220703125, 6103515625, 30517578125,
152587890625, 152586328121, 152578515601,
152539453001, 152344140001, 151367575001,
146484750001, 122070625001, 1 }
For t=9 the 27 elements in NPh27 are not tabulated here
For t=10 the 30 elements in NPh30 are not tabulated here
For t=11 the 33 elements in NPh33 are not tabulated here
For t=12 the 36 elements in NPh36 are not tabulated here
For t=13 the 39 elements in NPh39 are not tabulated here
For t=14 the 42 elements in NPh42 are not tabulated here
For t=15 the 45 elements in NPh45 are not tabulated here
For t=16 the 48 elements in NPh48 are not tabulated here
For t=17 the 51 elements in NPh51 are not tabulated here
For t=18 the 54 elements in NPh54 are not tabulated here
For t=19 the 57 elements in NPh57 are not tabulated here
For t=20 the 60 elements in NPh60 are not tabulated here
Using sturdy element notation we might say the results tabulated in this post are
