Yes! If a is a primitve root modulo p^n for n \geq 2,n \in \mathbb{Z},p \neq 2, then a is a primitive root modulo p^{n+1}. So, 10 is a primitive root modulo every prime power of 17 and 19. If p is a prime with p \nmid 10 and 10 is a primitive root modulo p^k for every k \in \mathbb{N}, then \phi(p^k)=p^k-p^{k-1}=p^{k-1}(p-1). When we plot these blue points on the graph, they will be points of the form (p^k,p^k-p^{k-1}). The slope of the line on which these points lie is:
I believe there is a real consequence to this. Our good numbers (previously defined) which are full reptend primes all lie on a line of slope 1>1-\frac{1}{p} for every p \in \mathbb{N}.If in fact there exist infinitely many full reptend primes (a fact which has not been proven, but which has been proven under the assumption of a generalized Riemman hypothesis, and which is believed to be true), then for a given fixed prime p=p_i, only finitely many numbers of the form p_i^k are good numbers.
I’m not sure if we both understand each other here but at the moment the program I’ve written is as fast as I can make it and need it to be (although I’m sure it can be faster).
For the scope of this report, I’m just trying to reduce the list of numbers that the program needs to check for a maximal period decimal, and based on some previous posts by @RayMan I assumed that we could discard any numbers not of the form p^k.
That means what I’m trying to do at the moment is justify that assumption mathematically although I’m getting slightly confused right now as to if the assumption was correct or not.
With regards to this question, I ran the program again for n=50000 and these are all the blue points above the line with slope 1/2.
x
y
fac
49
42
{7: 2}
289
272
{17: 2}
343
294
{7: 3}
361
342
{19: 2}
529
506
{23: 2}
841
812
{29: 2}
2209
2162
{47: 2}
2401
2058
{7: 4}
3481
3422
{59: 2}
3721
3660
{61: 2}
4913
4624
{17: 3}
6859
6498
{19: 3}
9409
9312
{97: 2}
11881
11772
{109: 2}
12167
11638
{23: 3}
12769
12656
{113: 2}
16807
14406
{7: 5}
17161
17030
{131: 2}
22201
22052
{149: 2}
24389
23548
{29: 3}
27889
27722
{167: 2}
32041
31862
{179: 2}
32761
32580
{181: 2}
37249
37056
{193: 2}
49729
49506
{223: 2}
For this range, all blue points above the 1/2 line / close to the red line are all prime powers.
I may have missed the explanation / not read properly but all these prime powers have not just prime bases but full repetend prime bases?