A Congruent Property of Gibonacci Number Modulo Prime

Abstract

Let a, b ∈ Z and p be a prime number such that a and b are not divisible by p. In this work, we give a congruent property modulo a prime number p of the gibonacci number defined by G_n = G_n−1 + G_n−2 with initial condition G₁ = a, G₂ = b. We show that a the gibonacci sequence satisfying G_kp−(p/5) ≡ G_k−1 (mod p) for all positive integer k and such odd prime p ≠ 5 if and only if a ≡ b (mod p). Moreover, for each odd prime number p, we give a necessary and sufficient condition yielding G_kp−(p/5) ≡ G_k−1 (mod p). We also find a relation between the sequences in the same equivalent class in modulo 5 constructed by Aoki and Sakai [1] that leads to such congruent property.