Research Article
BibTex RIS Cite
Year 2023, Issue: 45, 73 - 82, 31.12.2023
https://doi.org/10.53570/jnt.1368751

Abstract

References

  • T. Koshy, Fibonacci and Lucas Numbers with Applications, John Wiley and Sons, New York, 2001.
  • N. J. A. Sloane, The On-Line Encyclopedia of Integer Sequences (1964), https://oeis.org/, Accessed 20 Sep 2023.
  • T. Koshy, Elementary Number Theory with Applications, 2nd Edition, Academic Press, California, 2007.
  • U. Dudley, B. Tucker, Greatest Common Divisors in Altered Fibonacci Sequences, Fibonacci Quarterly 9 (1971) 89–91.
  • S. Hernandez, F. Luca, Common Factors of Shifted Fibonacci Numbers, Periodica Mathematica Hungarica 47 (2003) 95–110.
  • J. Spilker, The GCD of the Shifted Fibonacci Sequence, in: J. Sander, J. Steuding, R. Steuding (Eds.), From Arithmetic to Zeta-Functions: Number Theory in Memory of Wolfgang Schwarz, Springer, Cham, 2016, pp. 473–483.
  • Chen, K.W, Greatest Common Divisors in Shifted Fibonacci Sequences, Journal of Integer Sequences 14 (11) (2011) 4–7.
  • F. Koken, The GCD Sequences of the Altered Lucas Sequences, Annales Mathematicae Silesianae 34 (2) (2020) 222–240.
  • N. Robbins, Fibonacci and Lucas numbers of the Forms $w^2-1$, $w^3±1$, Fibonacci Quarterly 19 (4) (1981) 369–373.
  • J. H. E. Cohn, Square Fibonacci Numbers, The Fibonacci Quarterly 2 (2) (1964) 109–113.
  • H. London, R. Finkelstein, On Fibonacci and Lucas Numbers Which are Perfect Powers, The Fibonacci Quarterly 7 (5) (1969) 476–481.
  • R. Finkelstein, On Lucas Numbers Which are One More Than a Square, Fibonacci Quarterly 14 (1) (1973) 340–342.
  • H. C. Williams, On Fibonacci Numbers of the Form $k^2+1$, The Fibonacci Quarterly 13 (2) (1975) 213–214.
  • J. C. Lagarias, D. P. Weisser, Fibonacci and Lucas Cubes, The Fibonacci Quarterly 19 (1) (1981) 39–43.
  • D. Marques, The Fibonacci Version of the Brocard–Ramanujan Diophantine Equation, Portugaliae Mathematica 68 (2) (2011) 185–189.
  • L. Szalay, Diophantine Equations with Binary Recurrences Associated to the Brocard–Ramanujan Problem, Portugaliae Mathematica 69 (3) (2012) 213–220.
  • P. Pongsriiam, Fibonacci and Lucas Numbers Associated with Brocard-Ramanujan Equation, Communications of the Korean Mathematical Society 32 (3) (2017) 511–522.
  • Z. Cerin, On Factors of Sums of Consecutive Fibonacci and Lucas Numbers, Annales Mathematicae et Informaticae 41 (2013) 19–25.
  • A. Tekcan, A. Ozkoc, B. Gezer, O. Bizim, Some Relations Involving the Sums of Fibonacci Numbers, Proceedings of the Jangjeon Mathematical Society 11 (1) (2008) 1–12.
  • F. Koken, E. Kankal, Altered Numbers of Lucas Number Squared, Journal of Scientific Reports A 54 (2023) 62–75.

Altered Numbers of Fibonacci Number Squared

Year 2023, Issue: 45, 73 - 82, 31.12.2023
https://doi.org/10.53570/jnt.1368751

Abstract

We investigate two types of altered Fibonacci numbers obtained by adding or subtracting a specific value $\{a\}$ from the square of the $n^{th}$ Fibonacci numbers $G^{(2)}_{F(n)}(a)$ and $H^{(2)}_{F(n)}(a)$. These numbers are significant as they are related to the consecutive products of the Fibonacci numbers. As a result, we establish consecutive sum-subtraction relations of altered Fibonacci numbers and their Binet-like formulas. Moreover, we explore greatest common divisor (GCD) sequences of r-successive terms of altered Fibonacci numbers represented by $\left\{G^{(2)}_{F(n), r}(a)\right\}$ and $\left\{H^{(2)}_{F(n), r}(a)\right\}$ such that $r\in\{1,2,3\}$ and $a\in\{1,4\}$. The sequences are based on the GCD properties of consecutive terms of the Fibonacci numbers and structured as periodic or Fibonacci sequences.

References

  • T. Koshy, Fibonacci and Lucas Numbers with Applications, John Wiley and Sons, New York, 2001.
  • N. J. A. Sloane, The On-Line Encyclopedia of Integer Sequences (1964), https://oeis.org/, Accessed 20 Sep 2023.
  • T. Koshy, Elementary Number Theory with Applications, 2nd Edition, Academic Press, California, 2007.
  • U. Dudley, B. Tucker, Greatest Common Divisors in Altered Fibonacci Sequences, Fibonacci Quarterly 9 (1971) 89–91.
  • S. Hernandez, F. Luca, Common Factors of Shifted Fibonacci Numbers, Periodica Mathematica Hungarica 47 (2003) 95–110.
  • J. Spilker, The GCD of the Shifted Fibonacci Sequence, in: J. Sander, J. Steuding, R. Steuding (Eds.), From Arithmetic to Zeta-Functions: Number Theory in Memory of Wolfgang Schwarz, Springer, Cham, 2016, pp. 473–483.
  • Chen, K.W, Greatest Common Divisors in Shifted Fibonacci Sequences, Journal of Integer Sequences 14 (11) (2011) 4–7.
  • F. Koken, The GCD Sequences of the Altered Lucas Sequences, Annales Mathematicae Silesianae 34 (2) (2020) 222–240.
  • N. Robbins, Fibonacci and Lucas numbers of the Forms $w^2-1$, $w^3±1$, Fibonacci Quarterly 19 (4) (1981) 369–373.
  • J. H. E. Cohn, Square Fibonacci Numbers, The Fibonacci Quarterly 2 (2) (1964) 109–113.
  • H. London, R. Finkelstein, On Fibonacci and Lucas Numbers Which are Perfect Powers, The Fibonacci Quarterly 7 (5) (1969) 476–481.
  • R. Finkelstein, On Lucas Numbers Which are One More Than a Square, Fibonacci Quarterly 14 (1) (1973) 340–342.
  • H. C. Williams, On Fibonacci Numbers of the Form $k^2+1$, The Fibonacci Quarterly 13 (2) (1975) 213–214.
  • J. C. Lagarias, D. P. Weisser, Fibonacci and Lucas Cubes, The Fibonacci Quarterly 19 (1) (1981) 39–43.
  • D. Marques, The Fibonacci Version of the Brocard–Ramanujan Diophantine Equation, Portugaliae Mathematica 68 (2) (2011) 185–189.
  • L. Szalay, Diophantine Equations with Binary Recurrences Associated to the Brocard–Ramanujan Problem, Portugaliae Mathematica 69 (3) (2012) 213–220.
  • P. Pongsriiam, Fibonacci and Lucas Numbers Associated with Brocard-Ramanujan Equation, Communications of the Korean Mathematical Society 32 (3) (2017) 511–522.
  • Z. Cerin, On Factors of Sums of Consecutive Fibonacci and Lucas Numbers, Annales Mathematicae et Informaticae 41 (2013) 19–25.
  • A. Tekcan, A. Ozkoc, B. Gezer, O. Bizim, Some Relations Involving the Sums of Fibonacci Numbers, Proceedings of the Jangjeon Mathematical Society 11 (1) (2008) 1–12.
  • F. Koken, E. Kankal, Altered Numbers of Lucas Number Squared, Journal of Scientific Reports A 54 (2023) 62–75.
There are 20 citations in total.

Details

Primary Language English
Subjects Algebra and Number Theory
Journal Section Research Article
Authors

Fikri Köken 0000-0002-8304-9525

Emre Kankal 0000-0002-2707-5323

Early Pub Date December 30, 2023
Publication Date December 31, 2023
Submission Date September 29, 2023
Published in Issue Year 2023 Issue: 45

Cite

APA Köken, F., & Kankal, E. (2023). Altered Numbers of Fibonacci Number Squared. Journal of New Theory(45), 73-82. https://doi.org/10.53570/jnt.1368751
AMA Köken F, Kankal E. Altered Numbers of Fibonacci Number Squared. JNT. December 2023;(45):73-82. doi:10.53570/jnt.1368751
Chicago Köken, Fikri, and Emre Kankal. “Altered Numbers of Fibonacci Number Squared”. Journal of New Theory, no. 45 (December 2023): 73-82. https://doi.org/10.53570/jnt.1368751.
EndNote Köken F, Kankal E (December 1, 2023) Altered Numbers of Fibonacci Number Squared. Journal of New Theory 45 73–82.
IEEE F. Köken and E. Kankal, “Altered Numbers of Fibonacci Number Squared”, JNT, no. 45, pp. 73–82, December 2023, doi: 10.53570/jnt.1368751.
ISNAD Köken, Fikri - Kankal, Emre. “Altered Numbers of Fibonacci Number Squared”. Journal of New Theory 45 (December 2023), 73-82. https://doi.org/10.53570/jnt.1368751.
JAMA Köken F, Kankal E. Altered Numbers of Fibonacci Number Squared. JNT. 2023;:73–82.
MLA Köken, Fikri and Emre Kankal. “Altered Numbers of Fibonacci Number Squared”. Journal of New Theory, no. 45, 2023, pp. 73-82, doi:10.53570/jnt.1368751.
Vancouver Köken F, Kankal E. Altered Numbers of Fibonacci Number Squared. JNT. 2023(45):73-82.


TR Dizin 26024

Electronic Journals Library (EZB) 13651



Academindex 28993

SOBİAD 30256                                                   

Scilit 20865                                                  


29324 As of 2021, JNT is licensed under a Creative Commons Attribution-NonCommercial 4.0 International Licence (CC BY-NC).