A339517 Odd composite integers m such that A000032(2*m-J(m,5)) == J(m,5) (mod m), where J(m,5) is the Jacobi symbol.

323, 377, 1001, 1183, 1729, 1891, 3827, 4181, 5777, 6601, 6721, 8149, 8841, 10877, 11663, 13201, 13981, 15251, 17119, 17711, 18407, 19043, 23407, 25877, 26011, 27323, 30889, 34561, 34943, 35207, 39203, 40501, 41041

Offset: 1

Author

Ovidiu Bagdasar, Dec 07 2020

Keywords

• nonn

Comments

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=2, while V(m) recovers A000032(m) (Lucas numbers).

References

• 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).

Links

• Dorin Andrica and Ovidiu Bagdasar, On Generalized Lucas Pseudoprimality of Level k, Mathematics (2021) Vol. 9, 838.
• Dorin Andrica, Vlad Crişan, and Fawzi Al-Thukair, On Fibonacci and Lucas sequences modulo a prime and primality testing, Arab Journal of Mathematical Sciences, 24(1), 9-15 (2018).

Crossrefs

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

Cf. A339518 (a=3, b=-1), A339519 (a=5, b=-1), A339520 (a=7, b=-1).

Programs

Mathematica

Select[Range[3, 45000, 2], CoprimeQ[#, 5] && CompositeQ[#] && Divisible[LucasL[2*# - JacobiSymbol[#, 5]] - JacobiSymbol[#, 5], #] &]