A Congruent Property of Gibonacci Number Modulo Prime

Main Article Content

Wipawee Tangjai, Kodchaphon Wanichang, Montathip Srikao, Punyanuch Kheawkrai


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 [1] that leads to such congruent property.

Article Details


  1. M. Aoki, Y. Sakai, On Divisibility of Generalized Fibonacci Number, Integers, 15 (2015), A31.
  2. T. Koshy, Fibonacci and Lucas Numbers with Applications, John Wiley & Sons, Ltd, New Jersey, (2017).
  3. D. Andrica, V. Crişan, F. Al-Thukair, On Fibonacci and Lucas Sequences Modulo a Prime and Primality Testing, Arab J. Math. Sci. 24 (2018), 9–15. https://doi.org/10.1016/j.ajmsc.2017.06.002.
  4. J.D. Fulton, W.L. Morris, On Arithmetical Functions Related to the Fibonacci Numbers, Acta Arithmetica, 16 (1969), 105–110.
  5. S. Gupta, P. Rockstroh, F.E. Su, Splitting Fields and Periods of Fibonacci Sequences Modulo Primes, Math. Mag. 85 (2012), 130-135. https://doi.org/10.4169/math.mag.85.2.130.
  6. H. London, Fibonacci and Lucas Numbers, by Verner E. Hoggatt Jr. Houghton Mifflin Company, Boston, 1969. Canadian Math. Bull. 12 (1969), 367–367. https://doi.org/10.1017/S0008439500030514.
  7. N.J.A. Sloan, The On-Line Encyclopedia of Integer Sequences, http://oeis.org/A001175.
  8. J. Sondow, The On-Line Encyclopedia of Integer Sequences, https://oeis.org/A237437.