Mathematica Bohemica, Vol. 145, No. 1, pp. 19-32, 2020
Fermat $k$-Fibonacci and $k$-Lucas numbers
Jhon J. Bravo, Jose L. Herrera
Received February 6, 2018. Published online November 27, 2018.
Abstract: Using the lower bound of linear forms in logarithms of Matveev and the theory of continued fractions by means of a variation of a result of Dujella and Pethő, we find all $k$-Fibonacci and $k$-Lucas numbers which are Fermat numbers. Some more general results are given.
Keywords: generalized Fibonacci number; Fermat number, linear form in logarithms; reduction method
References: [1] E. F. Bravo, J. J. Bravo, F. Luca: Coincidences in generalized Lucas sequences. Fibonacci Q. 52 (2014), 296-306. MR 3276052 | Zbl 1375.11014
[2] J. J. Bravo, C. A. Gómez, F. Luca: Powers of two as sums of two $k$-Fibonacci numbers. Miskolc Math. Notes 17 (2016), 85-100. DOI 10.18514/MMN.2016.1505 | MR 3527869 | Zbl 1389.11041
[3] J. J. Bravo, F. Luca: Powers of two in generalized Fibonacci sequences. Rev. Colomb. Mat. 46 (2012), 67-79. MR 2945671 | Zbl 1353.11020
[4] J. J. Bravo, F. Luca: On a conjecture about repdigits in $k$-generalized Fibonacci sequences. Publ. Math. 82 (2013), 623-639. DOI 10.5486/PMD.2013.5390 | MR 3066434 | Zbl 1274.11035
[5] J. J. Bravo, F. Luca: Repdigits in $k$-Lucas sequences. Proc. Indian Acad. Sci., Math. Sci. 124 (2014), 141-154. DOI 10.1007/s12044-014-0174-7 | MR 3218885 | Zbl 1382.11019
[6] G. P. Dresden, Z. Du: A simplified Binet formula for $k$-generalized Fibonacci numbers. J. Integer Seq. 17 (2014), Article No. 14.4.7, 9 pages. MR 3181762 | Zbl 1360.11031
[7] A. Dujella, A. Pethő: A generalization of a theorem of Baker and Davenport. Quart. J. Math., Oxf. II. Ser. 49 (1998), 291-306. DOI 10.1093/qjmath/49.195.291 | MR 1645552 | Zbl 0911.11018
[8] R. Finkelstein: On Fibonacci numbers which are more than a square. J. Reine Angew. Math. 262/263 (1973), 171-178. DOI 10.1515/crll.1973.262-263.171 | MR 0325511 | Zbl 0265.10008
[9] R. Finkelstein: On Lucas numbers which are one more than a square. Fibonacci Q. 136 (1975), 340-342. MR 0422134 | Zbl 0319.10011
[10] L. K. Hua, Y. Wang: Applications of Number Theory to Numerical Analysis. Springer, Berlin; Science Press, Beijing (1981). MR 0617192 | Zbl 0451.10001
[11] F. Luca: Fibonacci and Lucas numbers with only one distinct digit. Port. Math. 57 (2000), 243-254. MR 1759818 | Zbl 0958.11007
[12] D. Marques: On $k$-generalized Fibonacci numbers with only one distinct digit. Util. Math. 98 (2015), 23-31. MR 3410879 | Zbl 1369.11014
[13] E. M. Matveev: An explicit lower bound for a homogeneous rational linear form in the logarithms of algebraic numbers II. Izv. Math. 64 (2000), 1217-1269; translated from Izv. Ross. Akad. Nauk Ser. Mat. 64 (2000), 125-180. DOI 10.1070/IM2000v064n06ABEH000314 | MR 1817252 | Zbl 1013.11043
[14] T. D. Noe, J. Vos Post: Primes in Fibonacci $n$-step and Lucas $n$-step sequences. J. Integer Seq. 8 (2005), Article No. 05.4.4, 12 pages. MR 2165333 | Zbl 1101.11008
[15] D. A. Wolfram: Solving generalized Fibonacci recurrences. Fibonacci Q. 36 (1998), 129-145. MR 1622060 | Zbl 0911.11014
