|
%I #8 Dec 16 2020 05:53:12
%S 9,25,27,33,35,45,65,81,99,117,121,161,175,221,225,297,325,363,585,
%T 645,705,825,891,1089,1281,1539,1541,1881,2025,2133,2145,2181,2299,
%U 2325,2925,3025,3267,3605,3745,4181,4573,4579,5265,5633,6721,6993,7245,7425,7865
%N Odd composite integers m such that A087130(3*m-J(m,29)) == 27*J(m,29) (mod m), where J(m,29) is the Jacobi symbol.
%C 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.
%C 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.
%C Here b=-1, a=5, D=29 and k=3, while V(m) recovers A087130(m), with V(2)=27.
%D D. Andrica, O. Bagdasar, Recurrent Sequences: Key Results, Applications and Problems. Springer, 2020.
%D D. Andrica, O. Bagdasar, On some new arithmetic properties of the generalized Lucas sequences, Mediterr. J. Math. (to appear, 2021).
%D D. Andrica, O. Bagdasar, On generalized pseudoprimality of level k (submitted).
%H 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).
%t Select[Range[3, 8000, 2], CoprimeQ[#, 29] && CompositeQ[#] && Divisible[LucasL[3*# - JacobiSymbol[#, 29], 5] - 27*JacobiSymbol[#, 29], #] &]
%Y Cf. A087130, A071904, A339127 (a=5, b=-1, k=1), A339519 (a=5, b=-1, k=2).
%Y Cf. A339724 (a=1, b=-1), A339725 (a=3, b=-1), A339727 (a=7, b=-1).
%K nonn
%O 1,1
%A _Ovidiu Bagdasar_, Dec 14 2020
|
