Tribonacci numbers as sum or difference of powers of 2

Yıl 2025, Cilt: 21 Sayı: 2, 147 - 151, 27.06.2025

Fatih Erduvan

https://doi.org/10.18466/cbayarfbe.1528991

Öz

This paper investigates Tribonacci numbers can be expressed as either the sum or difference of two distinct powers of 2. Namely, we address the problem of expressing Tribonacci numbers in the form

T_n=2^x±2^y

in positive integers with 1≤y≤x. Our findings reveal specific instances where such representations are possible, including examples like the seventh Tribonacci number expressed both as the sum and the difference of powers of 2. Additionally, we identify Tribonacci numbers that can be represented as the differences of Mersenne numbers, specifically, the numbers 2, 4, 24, and 504. These results enhance the understanding of the structural properties of Tribonacci sequences and their relationships with exponential and Mersenne-based number systems.

Anahtar Kelimeler

Diophantine equations , Tribonacci numbers , Baker’s Theory

Kaynakça

• [1]. Bravo, JJ, Luca, F. 2014. Powers of two as sums of two Lucas numbers. Journal of Integer Sequences; 17(1): article no 14.8.3.
• [2]. Bravo, EF, Bravo, JJ. 2015. Powers of two as sums of three Fibonacci numbers. Lithuanian Mathematical Journal; 55(3). 301-311.
• [3]. Bravo, JJ, Gómez, CA, Luca, F. 2016. Powers of two as sums of two k-Fibonacci numbers. Miskolc Mathematical Notes; 17(1): 85-100.
• [4]. Bravo, JJ, Faye, B, Luca, F. 2017. Powers of two as sums of three Pell numbers. Taiwanese Journal of Mathematics; 21(4): 739-751.
• [5]. Erduvan, F, Keskin, R. 2019. Non-negative integer solutions of the equation F_n-F_m=5^a. Turkish Journal of Mathematics; 43(3): 1115-1123.
• [6]. Demirtürk, B. 2019. On the Diophantine equation L_n-L_m=2∙3^a. Periodica Mathematica Hungaricas; 79(2): 210-217.
• [7]. Tiebekabe, P, Diouf, I. 2021. Powers of three as difference of two Fibonacci numbers. JP Journal of Algebra, Number Theory and Applications; 49(2): 185-196.
• [8]. Aboudja, H., Hernane, M, Rihane, SE, Togbé, A. 2021. On perfect powers that are sums of two Pell numbers. Periodica Mathematica Hungarica; 82: 11-15.
• [9]. Şiar, Z, Keskin, R. 2020. On the Diophantine equation F_n-F_m=2^a. Colloquium Mathematicum; 159: 119-126.
• [10]. Irmak, N, Szalay, L. 2016. Tribonacci Numbers Close to the Sum 2^a+3^b+5^c. Mathematica Scandinavica; 118: 27-32.
• [11]. Y. Bugeaud, 2018. ‘‘Linear Forms in Logarithms and Applications’’ IRMA Lectures in Mathematics and Theoretical Physics 28, Zurich, European Mathematical Society, 1-176.
• [12]. Y. Bugeaud, M. Mignotte S. Siksek, 2006. ‘‘Classical and modular approaches to exponential Diophantine equations I. Fibonacci and Lucas perfect powers’’ Annals of Mathematics, 163(3), 969-1018.
• [13]. J.J. Bravo, C.A. Gomez, F. Luca, 2016. ‘‘Powers of two as sums of two k-Fibonacci numbers’’ Miskolc Mathematical Notes, 17(1), 85-100.
• [14]. Bravo, J.J., Luca, F. (2012). Powers of two in generalized Fibonacci sequences. Revista Colombiana de Matemáticas, 46(1), 67-79.