On square Tribonacci Lucas numbers

Year 2021, Volume: 50 Issue: 6, 1652 - 1657, 14.12.2021

Nurettin Irmak

https://doi.org/10.15672/hujms.651786

https://izlik.org/JA59XX23TD

Abstract

The Tribonacci-Lucas sequence $\left\{S_n\right\}$ is defined by the recurrence relation $S_{n+3}=S_{n+2}+S_{n+1}+S_n$ with $S_0=3,S_1=1,S_2=3.$ In this note, we show that $1$ is the only perfect square in Tribonacci-Lucas sequence for $n\equiv ̸1\left(\mathrm{mod}32\right)$ and $n\equiv ̸17\left(\mathrm{mod}96\right).$

Keywords

Tribonacci sequence , Tribonacci Lucas sequence , squares

References

• [1] B. U. Alfred, On square Lucas numbers, Fibonacci Quart. 2 (1), 11-12, 1964.
• [2] J. E. Cohn, Square fibonacci numbers, Fibonacci Quart. 2 (2), 109-113, 1964.
• [3] N. Irmak and M. Alp, Tribonacci numbers with indices in arithmetic progression and their sums, Miskolc Math. Notes 14 (1), 125-133, 2013.
• [4] A. Pethő, Perfect powers in second order recurrences, Topics in Classical Number Theory, Akadémiai Kiadó, Budapest, 1217-1227, 1981.
• [5] ——, Fifteen problems in number theory, Acta Univ. Sapientiae Math. 2 (1), 72-83, 2010.
• [6] N. Robbins, On Fibonacci numbers of the form $px^2$, where $p$ is prime, Fibonacci Quart. 21, 266-271, 1983.
• [7] ——, On Pell numbers of the form $Px^3$, where $P$ is prime, Fibonacci Quart. 22, 340-348, 1984.
• [8] O. Wylie, In the Fibonacci series $F_1 = 1$, $F_2 = 1$, $F_{n+1} = F_n + F_{n-1}$ the first, second and twelvth terms are squares, Amer. Math. Monthly 71, 220-222, 1964.