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

1,3

Comments

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

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.

Example

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.

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}]

Crossrefs

Cf. A000040, A000045, A241568.

Sequence in context: A242872 A329362 A372971 * A282900 A126014 A317420

Adjacent sequences: A241601 A241602 A241603 * A241605 A241606 A241607

Keyword

nonn

Author

Zhi-Wei Sun, Apr 26 2014

Status

approved