Fermat k -Fibonacci and k -Lucas numbers

Jhon J. Bravo; Jose L. Herrera

Mathematica Bohemica (2020)

  • Volume: 145, Issue: 1, page 19-32
  • ISSN: 0862-7959

Abstract

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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.

How to cite

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Bravo, Jhon J., and Herrera, Jose L.. "Fermat $k$-Fibonacci and $k$-Lucas numbers." Mathematica Bohemica 145.1 (2020): 19-32. <http://eudml.org/doc/297334>.

@article{Bravo2020,
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.},
author = {Bravo, Jhon J., Herrera, Jose L.},
journal = {Mathematica Bohemica},
keywords = {generalized Fibonacci number; Fermat number; linear form in logarithms; reduction method},
language = {eng},
number = {1},
pages = {19-32},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Fermat $k$-Fibonacci and $k$-Lucas numbers},
url = {http://eudml.org/doc/297334},
volume = {145},
year = {2020},
}

TY - JOUR
AU - Bravo, Jhon J.
AU - Herrera, Jose L.
TI - Fermat $k$-Fibonacci and $k$-Lucas numbers
JO - Mathematica Bohemica
PY - 2020
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 145
IS - 1
SP - 19
EP - 32
AB - 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.
LA - eng
KW - generalized Fibonacci number; Fermat number; linear form in logarithms; reduction method
UR - http://eudml.org/doc/297334
ER -

References

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  2. Bravo, J. J., Gómez, C. A., Luca, F., 10.18514/MMN.2016.1505, Miskolc Math. Notes 17 (2016), 85-100. (2016) Zbl1389.11041MR3527869DOI10.18514/MMN.2016.1505
  3. Bravo, J. J., Luca, F., Powers of two in generalized Fibonacci sequences, Rev. Colomb. Mat. 46 (2012), 67-79. (2012) Zbl1353.11020MR2945671
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  5. Bravo, J. J., Luca, F., 10.1007/s12044-014-0174-7, Proc. Indian Acad. Sci., Math. Sci. 124 (2014), 141-154. (2014) Zbl1382.11019MR3218885DOI10.1007/s12044-014-0174-7
  6. Dresden, G. P., Du, Z., A simplified Binet formula for k -generalized Fibonacci numbers, J. Integer Seq. 17 (2014), Article No. 14.4.7, 9 pages. (2014) Zbl1360.11031MR3181762
  7. Dujella, A., Pethő, A., 10.1093/qjmath/49.195.291, Quart. J. Math., Oxf. II. Ser. 49 (1998), 291-306. (1998) Zbl0911.11018MR1645552DOI10.1093/qjmath/49.195.291
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  9. Finkelstein, R., On Lucas numbers which are one more than a square, Fibonacci Q. 136 (1975), 340-342. (1975) Zbl0319.10011MR0422134
  10. Hua, L. K., Wang, Y., Applications of Number Theory to Numerical Analysis, Springer, Berlin; Science Press, Beijing (1981). (1981) Zbl0451.10001MR0617192
  11. Luca, F., Fibonacci and Lucas numbers with only one distinct digit, Port. Math. 57 (2000), 243-254. (2000) Zbl0958.11007MR1759818
  12. Marques, D., On k -generalized Fibonacci numbers with only one distinct digit, Util. Math. 98 (2015), 23-31. (2015) Zbl1369.11014MR3410879
  13. Matveev, E. M., 10.1070/IM2000v064n06ABEH000314, Izv. Math. 64 (2000), 1217-1269 translated from Izv. Ross. Akad. Nauk Ser. Mat. 64 2000 125-180. (2000) Zbl1013.11043MR1817252DOI10.1070/IM2000v064n06ABEH000314
  14. Noe, T. D., Post, J. Vos, Primes in Fibonacci n -step and Lucas n -step sequences, J. Integer Seq. 8 (2005), Article No. 05.4.4, 12 pages. (2005) Zbl1101.11008MR2165333
  15. Wolfram, D. A., Solving generalized Fibonacci recurrences, Fibonacci Q. 36 (1998), 129-145. (1998) Zbl0911.11014MR1622060

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