A384490 Numbers m such that both roots of x^2 - x - 1 modulo m are primitive roots modulo m.

41, 61, 109, 149, 241, 269, 389, 409, 449, 569, 601, 641, 701, 821, 929, 1129, 1181, 1201, 1301, 1321, 1429, 1481, 1489, 1609, 1801, 1889, 1901, 1949, 2129, 2141, 2309, 2341, 2381, 2549, 2609, 2741, 2909, 3061, 3109, 3181, 3209, 3221, 3229, 3361, 3449, 3541

Offset: 1

Author

Jay Anderson, May 31 2025

Keywords

• nonn

Comments

Empirical observation: For each m in this sequence A001175(m) = m-1 and A015134(m) = m+2.

Examples

			For m = 41 the roots of x^2 - x - 1 (mod 41) are 7 and 35. 7 and 35 are both primitive roots modulo 41.
		

Links

• Wikipedia, Primitive root modulo n.

Crossrefs

Cf. A001175, A015134.

Programs

Mathematica

test[p_]:=Module[{inv2,sqr},If[JacobiSymbol[5,p]==1,inv2=ModularInverse[2,p]; sqr=PowerMod[5,1/2,p]; {MultiplicativeOrder[Mod[inv2*(sqr-1),p],p],MultiplicativeOrder[Mod[inv2*(-sqr-1),p],p]} == {p-1,p-1},False]]; Cases[Prime[Range[4, 5000]], ?(test[#] &)] (* _Shenghui Yang, Jun 01 2025 *)

PARI

{ forprime(p=2, 3600, s=polrootsmod(x^2 - x - 1,p);
 if( #s==2 && p-1==znorder(Mod(s[1],p)) && p-1==znorder(Mod(s[2],p)),
 print1(p,", "); ); ); } \\ Joerg Arndt, May 31 2025