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A293201 Primes p with a primitive root g such that g^3 = g^2 + g + 1 mod p.

%I A293201 #17 Jan 05 2025 19:51:41
%S A293201 7,11,13,17,19,41,47,53,73,107,131,139,149,163,167,199,227,257,263,
%T A293201 269,271,293,311,347,349,359,373,401,419,421,431,479,523,557,599,617,
%U A293201 683,701,757,761,769,809,827,863,877,907,911,929,937,953,991,1031,1033
%N A293201 Primes p with a primitive root g such that g^3 = g^2 + g + 1 mod p.
%C A293201 Since g^3 = g^2 + g + 1, we have g^4 = g^3 + g^2 + g, g^5 = g^4 + g^3 + g^2, g^6 = g^5 + g^4 + g^3, ..., g^(k+3) = g^(k+2) + g^(k+1) + g^k. Hence using g and g^2 we can compute all powers of the primitive root similar to A003147, where we have g^(k+2) = g^(k+1) + g^k (see the Shanks reference).
%H A293201 Robert Israel, <a href="/A293201/b293201.txt">Table of n, a(n) for n = 1..10000</a>
%H A293201 D. Shanks, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/10-2/shanks-a.pdf">Fibonacci primitive roots</a>, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/10-2/shanks-b.pdf">end of article</a>, Fib. Quart., 10 (1972), 163-168, 181.
%p A293201 filter:= proc(p) local x,r;
%p A293201   if not isprime(p) then return false fi;
%p A293201   for r in map(t -> rhs(op(t)), [msolve(x^3-x^2-x-1,p)]) do
%p A293201     if numtheory:-order(r,p) = p-1 then return true fi
%p A293201   od;
%p A293201   false
%p A293201 end proc:
%p A293201 select(filter, [seq(i,i=3..2000,2)]); # _Robert Israel_, Oct 02 2017
%t A293201 selQ[p_] := AnyTrue[PrimitiveRootList[p], Mod[#^3 - #^2 - # - 1, p] == 0&];
%t A293201 Select[Prime[Range[2, 200]], selQ] (* _Jean-François Alcover_, Jul 29 2020 *)
%o A293201 (PARI)
%o A293201 Z(r, p)=znorder(Mod(r, p))==p-1;  \\ whether r is a primitive root mod p
%o A293201 Y(p)=for(r=2,p-2,if( Z(r,p) && Mod(r^3-r^2-r-1,p)==0 , return(1))); 0; \\ test p
%o A293201 forprime(p=2,10^3,if(Y(p),print1(p,", ")) );
%Y A293201 Cf. A003147 (primitive root g such that g^2 = g + 1 mod p).
%Y A293201 Cf. A293200 (primitive root g such that g^3 = g + 1 mod p).
%K A293201 nonn
%O A293201 1,1
%A A293201 _Joerg Arndt_, Oct 02 2017