The proof relies on techniques involving linear forms in logarithms.
📌 3 Key Takeaways
🎯
Objective & Hypothesis:In the present paper, we identify all Jacobsthal numbers that may be expressed as a product of three Fibonacci numbers.
📊
Key Findings & Data:More precisely, our main result shows that the only solution to the Diophantine equation $$ F_k F_l F_m = J_n $$ for $2<k<l<m$ is expressed as $(k,l,m,n) = (5,7,8,12)$.
💡
Conclusion & Impact:The proof relies on techniques involving linear forms in logarithms.
In the present paper, we identify all Jacobsthal numbers that may be expressed as a product of three Fibonacci numbers. More precisely, our main result shows that the only solution to the Diophantine equation $$ F_k F_l F_m = J_n $$ for $2<k<l<m$ is expressed as $(k,l,m,n) = (5,7,8,12)$. The proof relies on techniques involving linear forms in logarithms.
Özet
Bu makalede, üç Fibonacci sayısının çarpımı olarak ifade edilebilecek tüm Jacobsthal sayılarını belirliyoruz. Daha kesin bir ifadeyle, ana sonucumuz, Diophantine denkleminin tek çözümünün FkFlFm = Jn 2 < k < l < m için (k, l, m, n) = (5, 7, 8, 12) olarak ifade edildiğini göstermektedir. İspat, logaritmalarda doğrusal formları içeren tekniklere dayanmaktadır.
Kim, D., Cihan, Z. P., Özkaya, Z. D., İnam, İ. (2026). On a Diophantine equation with Jacobsthal and Fibonacci numbers. Hacettepe Journal of Mathematics and Statistics, Volume: 55 Issue: 5. https://doi.org/10.15672/hujms.1823402
A. Baker, Linear forms in the logarithms of algebraic numbers I, II, III, Mathematika, 13, 204-216, 1966; 14, 102-107, 1967; 14, 220-228, 1967.
J. J. Bravo, C. A. Gómez and F. Luca, Powers of two as sums of two k-Fibonacci numbers, Miskolc Mathematical Notes, 17 (1), 85-100, 2016.
J. J. Bravo and F. Luca, Powers of two as sums of two Lucas numbers, Journal of Integer Sequences, 17 (8), 2014.
Y. Bugeaud, Linear forms in logarithms and applications, European Mathematical Society, 2018.
A. Dasdemir and A. Emin, Fibonacci and Lucas numbers as products of their arbitrary terms, Eskisehir Technical University Journal of Science and Technology A, 25 (3) 407-414, 2024.
M. Ddamulira, F. Luca and M. Rakotomalala, Fibonacci numbers which are products of two Pell numbers, Fibonacci Quarterly, 54 (1), 11-18, 2016.
A. Dujella and A. Petho, A generalization of a theorem of Baker and Davenport, Quarterly Journal of Mathematics, 49 (195), 291-306, 1998.
A. Emin and A. Dasdemir, Diophantine equations on cross-multiplicative forms of the Pell and Pell-Lucas numbers, Balıkesir Üniversitesi Fen Bilimleri Enstitüsü Dergisi, 27 (2), 464-474, 2025.
F. Erduvan and R. Keskin, Fibonacci numbers which are products of two Jacobsthal numbers, Tbilisi Mathematical Journal, 14 (2), 105-116, 2021.
F. Luca, J. Odjoumani and A. Togbé, Tribonacci numbers that are products of two Fibonacci numbers, Fibonacci Quarterly, 61 (4), 298-304, 2023.
F. Luca and V. Patel, On perfect powers that are sums of two Fibonacci numbers, Journal of Number Theory, 189, 90-96, 2018.
E. M. Matveev, An explicit lower bound for a homogeneous rational linear form in the logarithms of algebraic numbers II, Izvestiya: Mathematics, 64 (6), 1217-1269, 2000.
S. E. Rihane, On k-Fibonacci and k-Lucas numbers written as a product of two Pell numbers, Boletín de la Sociedad Matemática Mexicana, 30 (1), Article 20, 2024.
S. E. Rihane, Y. Akrour and A. El Habibi, Fibonacci numbers which are products of three Pell numbers and Pell numbers which are products of three Fibonacci numbers, Boletín de la Sociedad Matemática Mexicana, 26 (3), 895-910, 2020.
S. G. Sánchez and F. Luca, Linear combinations of factorials and S-units in a binary recurrence sequence, Annales Mathématiques du Québec, 38 (2), 169-188, 2014.
Z. Siar and R. Keskin, On the Diophantine equation $F_n - F_m = 2^a$, Colloquium Mathematicum, 159 (1), 119-126, 2020.