A337233 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.

35, 119, 169, 385, 741, 779, 899, 935, 961, 1105, 1121, 1189, 1443, 1479, 2001, 2419, 2555, 2915, 3059, 3107, 3383, 3605, 3689, 3741, 3781, 3827, 4199, 4795, 4879, 4901, 5719, 6061, 6083, 6215, 6265, 6441, 6479, 6601, 6895, 6929, 6931, 6965, 7055, 7107, 7801, 8119

Offset: 1

Author

Ovidiu Bagdasar, Aug 20 2020

Keywords

• nonn

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).

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

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

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

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).

Crossrefs

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).

Programs

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, #] &]