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Composite integers m such that P(m)^2 == 1 (mod m), where P(m) is the m-th Pell number A000129(m). Also, odd composite integers m such that U(m)^2 == 1 (mod m) and V(m) == 6 (mod m), where U(m)=A001109(m) and V(m)=A003499(m) are the m-th generalized Lucas and Pell-Lucas numbers of parameters a=6 and b=1, respectively.
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%I #17 Nov 23 2023 12:01:49

%S 35,119,169,385,741,779,899,935,961,1105,1121,1189,1443,1479,2001,

%T 2419,2555,2915,3059,3107,3383,3605,3689,3741,3781,3827,4199,4795,

%U 4879,4901,5719,6061,6083,6215,6265,6441,6479,6601,6895,6929,6931,6965,7055,7107,7801,8119

%N Composite integers m such that P(m)^2 == 1 (mod m), where P(m) is the m-th Pell number A000129(m). Also, odd composite integers m such that U(m)^2 == 1 (mod m) and V(m) == 6 (mod m), where U(m)=A001109(m) and V(m)=A003499(m) are the m-th generalized Lucas and Pell-Lucas numbers of parameters a=6 and b=1, respectively.

%C For a, b integers, the following sequences are defined:

%C generalized Lucas sequences by U(m+2)=a*U(m+1)-b*U(m) and U(0)=0, U(1)=1,

%C generalized Pell-Lucas sequences by V(m+2)=a*V(m+1)-b*V(m) and V(0)=2, V(1)=a.

%C In general, one has U^2(p) == 1 and V(p)==a (mod p) whenever p is prime and b=1, -1.

%C The composite numbers satisfying these congruences may be called weak generalized Lucas-Bruckner pseudoprimes of parameters a and b.

%C For a=2 and b=-1, U(m) recovers A000129(m) (Pell numbers).

%C For a=6 and b=1, we have U(m)=A001109(m) and V(m)=A003499(m).

%C This sequence contains the odd composite integers for which the congruence A000129(m)^2 == 1 (mod m) holds.

%C This is also the sequence of odd composite numbers satisfying the congruences A001109(m)^2 == 1 and A003499(m)==a (mod m).

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

%H D. Andrica and O. Bagdasar, <a href="https://repository.derby.ac.uk/item/92yqq/on-some-new-arithmetic-properties-of-the-generalized-lucas-sequences">On some new arithmetic properties of the generalized Lucas sequences</a>, preprint for Mediterr. J. Math. 18, 47 (2021).

%t Select[Range[3, 25000, 2], CompositeQ[#] && Divisible[Fibonacci[#, 2]*Fibonacci[#, 2] - 1, #] &]

%t Select[Range[3, 20000, 2], CompositeQ[#] && Divisible[2*ChebyshevT[#, 3] - 6, #] && Divisible[ChebyshevU[#-1, 3]*ChebyshevU[#-1, 3] - 1, #] &]

%Y Cf. A337231 (a=1, odd terms), A337232 (a=1, even terms), A337629 (a=6, b=-1), A337778 (a=4, b=1), A337779 (a=5, b=1).

%K nonn

%O 1,1

%A _Ovidiu Bagdasar_, Aug 20 2020