A Congruent Property of Gibonacci Number Modulo Prime
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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 Gn = Gn−1 + Gn−2 with initial condition G1 = a, G2 = b. We show that a the gibonacci sequence satisfying Gkp−(p/5) ≡ Gk−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 Gkp−(p/5) ≡ Gk−1 (mod p). We also find a relation between the sequences in the same equivalent class in modulo 5 constructed by Aoki and Sakai  that leads to such congruent property.
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