cp's OEIS Frontend

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.

%I A331506 #29 Jan 21 2020 12:31:26
%S A331506 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,
%T A331506 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,
%U A331506 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
%N 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.
%C A331506 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.
%C A331506 This implies that for each odd prime p there exists a Fibonacci number F(k) < p which is a quadratic nonresidue modulo p.
%C A331506 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.
%C A331506 We also have the following conjecture similar to Conjecture 1.
%C A331506 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.
%C A331506 This has been verified for all primes p < 10^9.
%H A331506 Zhi-Wei Sun, <a href="/A331506/b331506.txt">Table of n, a(n) for n = 1..10000</a>
%H A331506 Zhi-Wei Sun, <a href="http://arxiv.org/abs/1405.0290">New observations on primitive roots modulo primes</a>, arXiv preprint arXiv:1405.0290 [math.NT], 2014.
%H A331506 Zhi-Wei Sun, <a href="http://mathoverflow.net/questions/350730">Primitive roots modulo primes related to Fibonacci numbers or Lucas numbers</a>, Question 350730 at MathOverflow, Jan. 19, 2020.
%e A331506 a(85) = 15 with 15 = 3*5 = F(4)*F(5) a primitive root modulo prime(85) = 439.
%t A331506 p[n_]:=p[n]=Prime[n];
%t A331506 Dv[n_]:=Dv[n]=Divisors[p[n]-1];
%t A331506 ls={};
%t A331506 Do[If[Fibonacci[k]Fibonacci[m]<p[100],ls=Append[ls,Fibonacci[k]Fibonacci[m]]],{k,2,14},{m,2,k}]
%t A331506 LL:=LL=Sort[DeleteDuplicates[ls]];
%t A331506 ff[r_]:=ff[r]=LL[[r]];
%t A331506 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]
%Y A331506 Cf. A000032, A000040, A000045, A001918, A239957, A241476, A241504, A241568, A241604.
%K A331506 nonn
%O A331506 1,2
%A A331506 _Zhi-Wei Sun_, Jan 18 2020