A241604 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.

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, 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, 2, 5, 2, 2, 3, 13, 3, 2, 2, 5, 2, 2, 3, 13

Offset: 1

Author

Zhi-Wei Sun, Apr 26 2014

Keywords

• nonn

Comments

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

Examples

			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.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
• Z.-W. Sun, New observations on primitive roots modulo primes, arXiv preprint arXiv:1405.0290 [math.NT], 2014.

Crossrefs

Cf. A000040, A000045, A241568.

Programs

Mathematica

f[k_]:=Fibonacci[k]
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}]