A331506 Least primitive root g < prime(n) of the n-th prime with g a product of two Fibonacci numbers, or 0 if such a number g does not exist.

1, 2, 2, 3, 2, 2, 3, 2, 5, 2, 3, 2, 6, 3, 5, 2, 2, 2, 2, 13, 5, 3, 2, 3, 5, 2, 5, 2, 6, 3, 3, 2, 3, 2, 2, 6, 5, 2, 5, 2, 2, 2, 21, 5, 2, 3, 2, 3, 2, 6, 3, 13, 13, 6, 3, 5, 2, 6, 5, 3, 3, 2, 5, 34, 10, 2, 3, 10, 2, 2, 3, 13, 6, 2, 2, 5, 2, 5, 3, 21, 2, 2, 13, 5, 15, 2, 3, 13, 2, 3, 2, 13, 3, 2, 10, 5, 2, 3, 2, 2

Offset

1,2

Comments

Conjecture 1: a(n) > 0 for all n > 0. In other words, for each prime p there are two Fibonacci numbers F(k) and F(m) with F(k)*F(m) < p such that F(k)*F(m) is a primitive root modulo p.

This implies that for each odd prime p there exists a Fibonacci number F(k) < p which is a quadratic nonresidue modulo p.

It seems that Conjecture 1 can be strengthened as follows: For any prime p, there is a primitive root g < p modulo p such that g/F(2) = g or g/F(3) = g/2 or g/F(4) = g/3 is a Fibonacci number. We have verified this strong version for all primes p < 5*10^9.

We also have the following conjecture similar to Conjecture 1.

Conjecture 2. For any prime p, there are two Lucas numbers L(k) and L(m) with k >= m >= 0 and L(k)*L(m) < p such that L(k)*L(m) is a primitive root modulo p.

This has been verified for all primes p < 10^9.

Links

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Zhi-Wei Sun, New observations on primitive roots modulo primes, arXiv preprint arXiv:1405.0290 [math.NT], 2014.

Zhi-Wei Sun, Primitive roots modulo primes related to Fibonacci numbers or Lucas numbers, Question 350730 at MathOverflow, Jan. 19, 2020.

Example

a(85) = 15 with 15 = 3*5 = F(4)*F(5) a primitive root modulo prime(85) = 439.

Mathematica

p[n_]:=p[n]=Prime[n];
Dv[n_]:=Dv[n]=Divisors[p[n]-1];
ls={};
Do[If[Fibonacci[k]Fibonacci[m]<p[100], ls=Append[ls, Fibonacci[k]Fibonacci[m]]], {k, 2, 14}, {m, 2, k}]
LL:=LL=Sort[DeleteDuplicates[ls]];
ff[r_]:=ff[r]=LL[[r]];
tab={}; Do[r=1; Label[aa]; If[ff[r]>=p[n], tab=Append[tab, 0]; Goto[bb]]; Do[If[PowerMod[ff[r], Dv[n][[i]], p[n]]==1, r=r+1; Goto[aa]], {i, 1, Length[Dv[n]]-1}]; tab=Append[tab, ff[r]]; Label[bb], {n, 1, 100}]; Print[tab]

Crossrefs

Cf. A000032, A000040, A000045, A001918, A239957, A241476, A241504, A241568, A241604.

Sequence in context: A127810 A001918 A268616 * A002233 A241516 A273458

Adjacent sequences: A331503 A331504 A331505 * A331507 A331508 A331509

Keyword

nonn

Author

Zhi-Wei Sun, Jan 18 2020

Status

approved