A350121 Increasing sequence of primes p == 3 (mod 4) such that all of 2,3,5,...,prime(n) are primitive roots mod p.

3, 19, 907, 1747, 2083, 101467, 350443, 916507, 1014787, 6603283, 27068563, 45287587, 226432243, 243060283, 3946895803, 5571195667, 9259384843, 19633449763, 229012273627, 965558895907, 2793054173947, 5142304754563

Offset: 1

Author

Paul Vanderveen, Dec 15 2021

Keywords

• nonn
• more

Comments

It is possible, although rather unlikely, that any primes congruent to 3 (mod 4) will appear in A213052.

a(19) > 10^11.

Examples

			a(2) = 19 since 19 is the smallest prime (congruent to 3 (mod 4)) such that the first two primes (2 and 3) are primitive roots.
		

Crossrefs

Cf. A213052.

Programs

Mathematica

max=0;Do[n=Prime@i;If[Mod[n,4]==3,k=1;While[MultiplicativeOrder[Prime@k,n]==n-1,k++];If[k-1>max,Print@n;max++]],{i,10^6}] (* Giorgos Kalogeropoulos, Dec 17 2021 *)

PARI

N=10^10;
default(primelimit, N);
A=2;
{ forprime (p=3, N,
    if (p%4==3,
    q = 1;
    forprime (a=2, A,
        if ( znorder(Mod(a, p)) != p-1,  q=0; break() );
    );
    if ( q, A=nextprime(A+1); print1(p, ", ") );
    );
); }

Extensions

a(19) from Daniel Suteu, Dec 20 2021

a(20)-a(21) from Paul Vanderveen, May 08 2025