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

1,1

Comments

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

Links

Table of n, a(n) for n=1..46.

Wikipedia, Primitive root modulo n.

Example

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.

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 *)

Prog

(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

Crossrefs

Cf. A001175, A015134.

Sequence in context: A139952 A195573 A260554 * A031415 A325072 A089345

Adjacent sequences: A384487 A384488 A384489 * A384491 A384492 A384493

Keyword

nonn

Author

Jay Anderson, May 31 2025

Status

approved