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

Offset

1,1

Comments

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

a(19) > 10^11.

Links

Table of n, a(n) for n=1..19.

Example

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.

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 *)

Prog

(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, ", ") );
    );
); }

Crossrefs

Cf. A213052.

Sequence in context: A330040 A168591 A319190 * A014015 A114301 A258669

Adjacent sequences: A350118 A350119 A350120 * A350122 A350123 A350124

Keyword

nonn,more

Author

Paul Vanderveen, Dec 15 2021

Extensions

a(19) from Daniel Suteu, Dec 20 2021

Status

approved