Lucas sequences of the first and second kinds are, respectively, the integer sequences $ (U_n)_{n\geq0} $ and $ (V_n)_{n\geq0} $ depending on parameters $ a, b\in\mathbb{Z} $ and defined by the recurrence relations $ U_0 = 0 $, $ U_1 = 1 $, and $ U_n = aU_{n-1}+bU_{n-2} $ for $ n\geq2 $, $ V_0 = 2 $, $ V_1 = a $, and $ V_n = aV_{n-1}+bV_{n-2} $ for $ n\geq2 $. In this article, we obtain exact divisibility results concerning $ U_n^k $ and $ V_n^k $ for all positive integers $ n $ and $ k $. This and our previous article extend many results in the literature and complete a long investigation on this problem from 1970 to 2021.
Citation: Kritkhajohn Onphaeng, Prapanpong Pongsriiam. Exact divisibility by powers of the integers in the Lucas sequences of the first and second kinds[J]. AIMS Mathematics, 2021, 6(11): 11733-11748. doi: 10.3934/math.2021682
Lucas sequences of the first and second kinds are, respectively, the integer sequences $ (U_n)_{n\geq0} $ and $ (V_n)_{n\geq0} $ depending on parameters $ a, b\in\mathbb{Z} $ and defined by the recurrence relations $ U_0 = 0 $, $ U_1 = 1 $, and $ U_n = aU_{n-1}+bU_{n-2} $ for $ n\geq2 $, $ V_0 = 2 $, $ V_1 = a $, and $ V_n = aV_{n-1}+bV_{n-2} $ for $ n\geq2 $. In this article, we obtain exact divisibility results concerning $ U_n^k $ and $ V_n^k $ for all positive integers $ n $ and $ k $. This and our previous article extend many results in the literature and complete a long investigation on this problem from 1970 to 2021.
| [1] | A. Benjamin, J. Rouse, When does $F_m^L$ divide $F_n$? A combinatorial solution, Proceedings of the Eleventh International Conference on Fibonacci Numbers and Their Applications, 194, Congressus Numerantium, 2009, 53–58. |
| [2] |
Y. Bugeaud, M. Mignotte, S. Siksek, Classical and modular approaches to exponential Diophantine equations I. Fibonacci and Lucas perfect powers, Ann. Math., 163 (2006), 969–1018. doi: 10.4007/annals.2006.163.969
|
| [3] |
P. Cubre, J. Rouse, Divisibility properties of the Fibonacci entry point, Proc. Amer. Math. Soc., 142 (2014), 3771–3785. doi: 10.1090/S0002-9939-2014-12269-6
|
| [4] | V. E. Hoggatt Jr., M. Bicknell-Johnson, Divisibility by Fibonacci and Lucas squares, Fibonacci Quart., 15 (1977), 3–8. |
| [5] | Y. Matijasevich, Enumerable Sets are Diophantine, Proc. Academy Sci. USSR, 11 (1970), 354–358. |
| [6] |
Y. Matijasevich, My collaboration with Julia Robison, Math. Intell., 14 (1992), 38–45. doi: 10.1007/BF03024472
|
| [7] | Y. Matijasevich, Hilbert's Tenth Problem, MIT Press, 1996. |
| [8] |
K. Onphaeng, P. Pongsriiam, Exact divisibility by powers of the integers in the Lucas sequence of the first kind, AIMS Math., 5 (2020), 6739–6748. doi: 10.3934/math.2020433
|
| [9] | K. Onphaeng, P. Pongsriiam, Subsequences and divisibility by powers of the Fibonacci numbers, Fibonacci Quart., 52 (2014), 163–171. |
| [10] | K. Onphaeng, P. Pongsriiam, The converse of exact divisibility by powers of the Fibonacci and Lucas numbers, Fibonacci Quart., 56 (2018), 296–302. |
| [11] | C. Panraksa, A. Tangboonduangjit, $p$-adic valuation of Lucas iteration sequences, Fibonacci Quart., 56 (2018), 348–353. |
| [12] | A. Patra, G. K. Panda, T. Khemaratchatakumthorn, Exact divisibility by powers of the balancing and Lucas-balancing numbers, Fibonacci Quart., 59 (2021), 57–64. |
| [13] | P. Phunphayap, P. Pongsriiam, Explicit formulas for the $p$-adic valuations of Fibonomial coefficients, J. Integer Seq., 21 (2018), Article 18.3.1. |
| [14] |
P. Phunphayap, P. Pongsriiam, Explicit formulas for the $p$-adic valuations of Fibonomial coefficients II, AIMS Math., 5 (2020), 5685–5699. doi: 10.3934/math.2020364
|
| [15] | P. Pongsriiam, Exact divisibility by powers of the Fibonacci and Lucas numbers, J. Integer Seq., 17 (2014), Article 14.11.2. |
| [16] | P. Pongsriiam, Fibonacci and Lucas numbers associated with Brocard-Ramanujan equation, Commun. Korean Math. Soc., 32 (2017), 511–522. |
| [17] |
M. K. Sahukar, G. K. Panda, Diophantine equations with balancing-like sequences associated to Brocard-Ramanujan-type problem, Glas Mat., 54 (2019), 255–270. doi: 10.3336/gm.54.2.01
|
| [18] | C. Sanna, The $p$-adic valuation of Lucas sequences, Fibonacci Quart., 54 (2016), 118–124. |
| [19] | J. Seibert, P. Trojovský, On divisibility of a relation of the Fibonacci numbers, Int. J. Pure Appl. Math., 46 (2008), 443–448. |
| [20] |
C. L. Stewart, On divisors of Lucas and Lehmer numbers, Acta Math., 211 (2013), 291–314. doi: 10.1007/s11511-013-0105-y
|
Kritkhajohn Onphaeng, Prapanpong Pongsriiam. Exact divisibility by powers of the integers in the Lucas sequences of the first and second kinds[J]. AIMS Mathematics, 2021, 6(11): 11733-11748. doi: 10.3934/math.2021682
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