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Abstract
We study generalized Fibonacci sequences
F n + 1
=
P F n
−
Q F n − 1 with initial
values
F 0
= 0
and
F 1
= 1 .
Let
P , Q
be relatively prime nonzero integers such that
P 2
− 4 Q is not a perfect square.
We show that if
Q
=
± 1
then the sequence
{ F n } n = 0 ∞
misses a congruence class modulo every large enough prime. If
Q ≠
± 1 , we prove under the
GRH that the sequence
{ F n } n = 0 ∞
hits every congruence class modulo infinitely many primes.
Keywords
generalized Fibonacci sequence, complete sequence
Mathematical Subject Classification 2010
Primary: 11B39
Secondary: 11B50
Milestones
Received: 17 December 2018
Revised: 7 November 2019
Accepted: 22 November 2019
Published: 8 January 2020