A241604 - OEIS

%I #16 Aug 05 2019 03:02:53

%S 0,0,2,3,2,2,3,2,5,2,3,2,3,2,5,2,2,2,2,13,5,3,2,3,5,2,3,2,2,3,3,2,3,2,

%T 2,3,2,2,5,2,2,2,21,5,2,3,2,3,2,2,3,13,13,2,3,5,2,3,2,3,2,2,2,34,5,2,

%U 2,5,2,2,3,13,3,2,2,5,2,2,3,13

%N Least Fibonacci number smaller than prime(n)/2 which is a quadratic nonresidue modulo prime(n), or 0 if such a Fibonacci number does not exist.

%C According to the conjecture in A241568, a(n) should be positive for all n > 2.

%H Zhi-Wei Sun, <a href="/A241604/b241604.txt">Table of n, a(n) for n = 1..10000</a>

%H Z.-W. 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.

%e a(4) = 3 since the Fibonacci number F(4) = 3 < prime(4)/2 is a quadratic nonresidue modulo prime(4) = 7, but the Fibonacci numbers F(1) = F(2) = 1 and F(3) = 2 are quadratic residues modulo prime(4) = 7.

%t f[k_]:=Fibonacci[k]

%t Do[Do[If[f[k]>Prime[n]/2,Goto[bb]];If[JacobiSymbol[f[k],Prime[n]]==-1,Print[n," ",Fibonacci[k]];Goto[aa]];Continue,{k,1,(Prime[n]+1)/2}];Label[bb];Print[n," ",0];Label[aa];Continue,{n,1,80}]

%Y Cf. A000040, A000045, A241568.

%K nonn

%O 1,3

%A _Zhi-Wei Sun_, Apr 26 2014