As you may notice, A213052 contains primes mostly congruent to $1\pmod 4$ (in fact, all of the known ones are except $3$).

Consider the sequence of smallest primes $p_n$ such that $2,3,5,7,11,13,...$ (first $n$ primes) are all primitive roots $\pmod {p_n}$.

So clearly, primes $p = 3\pmod 4$ could be part of this sequence (it is fairly easy to construct such primes):

The smallest example of a prime $p=3\pmod 4$ where $2, 3, 5$ are primitive roots for example, is $907$, while without this restriction, it is $53$. This can be done for arbitrary first $n$ primes, with such a restriction, we have the following $p_n$ congruent to $3 \pmod 4$:

$3, 19, 907, 1747, 2083, 101467, 350443,...$

These are much larger than the original sequence:

$3, 5, 53, 173, 293, 2477, 9173$

So I suspect that if a prime congruent to $3 \pmod 4$ is in A213052, it is pretty rare, although how likely is it to occur?

  • 1
    $\begingroup$ Do you mean that the first $n$ primes must be primitive roots ? And we look at the smallest prime with this property and then the smallest of the form $4k+3$ ? $\endgroup$
    – Peter
    Dec 15, 2021 at 11:08
  • $\begingroup$ Aditionally, if you mention how such a prime can be constructed this might help to find out why primes of the form $4n+1$ are preferred. $\endgroup$
    – Peter
    Dec 15, 2021 at 11:50
  • 1
    $\begingroup$ I think it has something to do with $3$ being a Quadratic Non-Residue but it could be something else too. $\endgroup$
    – J. Linne
    Dec 15, 2021 at 18:13


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