OFFSET
1,1
COMMENTS
For a, b integers, the following sequences are defined:
generalized Lucas sequences by U(m+2)=a*U(m+1)-b*U(m) and U(0)=0, U(1)=1,
generalized Pell-Lucas sequences by V(m+2)=a*V(m+1)-b*V(m) and V(0)=2, V(1)=a.
In general, one has U^2(p) == 1 and V(p)==a (mod p) whenever p is prime and b=1, -1.
The composite numbers satisfying these congruences may be called weak generalized Lucas-Bruckner pseudoprimes of parameters a and b.
For a=2 and b=-1, U(m) recovers A000129(m) (Pell numbers).
This sequence contains the odd composite integers for which the congruence A000129(m)^2 == 1 (mod m) holds.
REFERENCES
D. Andrica, O. Bagdasar, Recurrent Sequences: Key Results, Applications and Problems. Springer, 2020.
LINKS
D. Andrica and O. Bagdasar, On some new arithmetic properties of the generalized Lucas sequences, preprint for Mediterr. J. Math. 18, 47 (2021).
MATHEMATICA
Select[Range[3, 25000, 2], CompositeQ[#] && Divisible[Fibonacci[#, 2]*Fibonacci[#, 2] - 1, #] &]
Select[Range[3, 20000, 2], CompositeQ[#] && Divisible[2*ChebyshevT[#, 3] - 6, #] && Divisible[ChebyshevU[#-1, 3]*ChebyshevU[#-1, 3] - 1, #] &]
CROSSREFS
KEYWORD
nonn
AUTHOR
Ovidiu Bagdasar, Aug 20 2020
STATUS
approved