Some Divisibility Properties of Lucas Numbers

Yıl 2021, Cilt: 13 Sayı: 2, 234 - 238, 31.12.2021

Adem Şahin Sadettin Karagöl

https://doi.org/10.47000/tjmcs.783597

Öz

The Lucas number sequence is a popular number sequence that has been described as similar to the Fibonacci number sequence. A lot of research has been done on this number sequence. Some of these studies are on the divisibility properties of this number sequence. Carlitz (1964) examined the requirement that a given Lucas number can be divided by another Lucas number. After that, many studies have been done on this subject. In the present article, we obtain some divisibility properties of the Lucas Numbers. First, we examine the case $L_{(2n-1)m}/L_{m}$ and then we obtain $L_{\left( 2n-1\right) m}$ using different forms of Lucas numbers.

Anahtar Kelimeler

Lucas sequence, divisibility, Recurrence relation

Kaynakça

• [1] Carlitz, L., A Note On Fibonacci numbers, Fibonacci Quarterly, 2(1)(1964), 15–28.
• [2] Carlitz, L., Hunter, J.A.H., Sum of powers of Fibonacci and Lucas numbers, Fibonacci Quarterly, 7(5)(1969), 467–473.
• [3] Carlitz, L., A Conjecture concerning Lucas numbers, Fibonacci Quarterly, 10(5)(1972), 526–550.
• [4] Di Porto, A., Filipponi, P., A Probabilistic Primality Test Based on the Properties of Certain Generalized Lucas Numbers. In: Barstow D. et al. (eds) Advances in Cryptology, EUROCRYPT 88. EUROCRYPT 1988. Lecture Notes in Computer Science, vol 330. Springer, Berlin, Heidelberg, 1988.
• [5] Hoggatt Jr., V.E., An application of the Lucas triangle, Fibonacci Quarterly, 8(4)(1970), 360–364.
• [6] Hoggatt Jr., V.E., Bergum, G.E., Divisibility and congruence relations, Fibonacci Quarterly, 12(2)(1974), 189–195.
• [7] Keskin, R., Demirturk Bitim, B., Fibonacci and Lucas congruences and their applications, Acta Math. Sinica, 27(4)(2011),725–736.
• [8] Koshy, T., New Fibonacci and Lucas identities, The Mathematical Gazette, 82(495)(1998), 481–484.
• [9] Koshy, T., The convergence of a Lucas series, The Mathematical Gazette, 83(497)(1999), 272–274.
• [10] Koshy, T., Fibonacci and Lucas Numbers With Aplications. John Wiley & Sons Inc., New York, USA, 2001.
• [11] Vajda, S., Fibonacci and Lucas Numbers and The Golden Section, Ellis Horwood Limited, Chichester, England, 1989.