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D. Andrica, V. Crisan, F. Al-Thukair, On Fibonacci and Lucas sequences modulo a prime and primality testing. Arab J. Math. Sci. 24(1), 9-15 (2018).

Dorin Andrica, Vlad Crişan, and Fawzi Al-Thukair, <a href="https://doi.org/10.1016/j.ajmsc.2017.06.002">On Fibonacci and Lucas sequences modulo a prime and primality testing</a>, Arab Journal of Mathematical Sciences, 24(1), 9-15 (2018).

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allocated for Ovidiu BagdasarOdd composite integers m such that A000032(3*m-J(m,5)) == 3*J(m,5) (mod m), where J(m,5) is the Jacobi symbol.

9, 21, 161, 341, 901, 1281, 1853, 3201, 4181, 5473, 5611, 5777, 6119, 6721, 9729, 10877, 11041, 12209, 12441, 13201, 14981, 15251, 16771, 17941, 20591, 20769, 20801, 23323, 25761, 27403, 27661, 28121, 28421, 29489, 33001, 34561, 38801, 39281, 41159, 42721

1,1

The generalized Pell-Lucas sequences of integer parameters (a,b) defined by V(m+2)=a*V(m+1)-b*V(m) and V(0)=2, V(1)=a, satisfy V(k*p-J(p,D)) == V(k-1)*J(p,D) (mod p) whenever p is prime, k is a positive integer, b=-1 and D=a^2+4.

The composite integers m with the property V(k*m-J(m,D)) == V(k-1)*J(m,D) (mod m) are called generalized Pell-Lucas pseudoprimes of level k- and parameter a.

Here b=-1, a=1, D=5 and k=3, while V(m) is A000032(m) (Lucas numbers), with V(2)=3.

D. Andrica, O. Bagdasar, Recurrent Sequences: Key Results, Applications and Problems. Springer, 2020.

D. Andrica, O. Bagdasar, On some new arithmetic properties of the generalized Lucas sequences, Mediterr. J. Math. (to appear, 2021).

D. Andrica, O. Bagdasar, On generalized pseudoprimality of level k (submitted).

D. Andrica, V. Crisan, F. Al-Thukair, On Fibonacci and Lucas sequences modulo a prime and primality testing. Arab J. Math. Sci. 24(1), 9-15 (2018).

Select[Range[3, 43000, 2], CoprimeQ[#, 5] && CompositeQ[#] && Divisible[LucasL[3*# - JacobiSymbol[#, 5]] - 3*JacobiSymbol[#, 5], #] &]

Cf. A000032, A071904, A339125 (a=1, b=-1, k=1), A339517 (a=1, b=-1, k=2).

Cf. A339725 (a=3, b=-1), A339726 (a=5, b=-1), A339727 (a=7, b=-1).

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Ovidiu Bagdasar, Dec 14 2020

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