A337231 Odd composite integers m such that F(m)^2 == 1 (mod m), where F(m) is the m-th Fibonacci number.

231, 323, 377, 1443, 1551, 1891, 2737, 2849, 3289, 3689, 3827, 4181, 4879, 5777, 6479, 6601, 6721, 7743, 8149, 9879, 10877, 11663, 13201, 13981, 15251, 15301, 17119, 17261, 17711, 18407, 19043, 20999, 23407, 25877, 27071, 27323, 29281, 30889, 34561, 34943, 35207

Offset: 1

Author

Ovidiu Bagdasar, Aug 20 2020

Keywords

• nonn

Comments

If p is a prime, then A000045(p)^2==1 (mod p).

This sequence contains the odd composite integers for which the congruence holds.

The generalized Lucas sequence of integer parameters (a,b) defined by U(n+2)=a*U(n+1)-b*U(n) and U(0)=0, U(1)=1, satisfies the identity U^2(p)==1 (mod p) whenever p is prime and b=-1.

For a=1, b=-1, U(n) recovers A000045(n) (Fibonacci numbers).

References

• D. Andrica, O. Bagdasar, Recurrent Sequences: Key Results, Applications and Problems. Springer (to appear, 2020).

Links

• Amiram Eldar, Table of n, a(n) for n = 1..1000
• Dorin Andrica and Ovidiu Bagdasar, On Generalized Lucas Pseudoprimality of Level k, Mathematics (2021) Vol. 9, 838.
• 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. A000045.

Programs

Mathematica

Select[Range[3, 30000, 2], CompositeQ[#] && Divisible[Fibonacci[#, 1]*Fibonacci[#, 1] - 1, #] &]

PARI

lista(nn) = my(list=List()); forcomposite(c=1, nn, if ((c%2) && (Mod(fibonacci(c), c)^2 == 1), listput(list, c))); Vec(list); \\ Michel Marcus, Sep 29 2023