## Abstract

For an integer k ≥ 2, let {F n } n≥0 be the k–generalized Fibonacci sequence

which starts with 0, . . . , 0, 1 (k terms) and each term afterwards is the sum of the k preceding terms. In this paper, we find all integers c having at least two representations as a difference between a k–generalized Fibonacci number and a power of 2 for any fixed k ≥ 4. This paper extends previous work from Ddamulira et al. (Proc Math Sci 127(3): 411–421, 2017. https://doi.org/10.100/s12044-017-0338-3) for the case k = 2 and Bravo et al. (Bull Korean Math Soc 54(3): 069–1080, 2017. https://doi.org/10.4134/BKMS.b160486) for the case k =3.

which starts with 0, . . . , 0, 1 (k terms) and each term afterwards is the sum of the k preceding terms. In this paper, we find all integers c having at least two representations as a difference between a k–generalized Fibonacci number and a power of 2 for any fixed k ≥ 4. This paper extends previous work from Ddamulira et al. (Proc Math Sci 127(3): 411–421, 2017. https://doi.org/10.100/s12044-017-0338-3) for the case k = 2 and Bravo et al. (Bull Korean Math Soc 54(3): 069–1080, 2017. https://doi.org/10.4134/BKMS.b160486) for the case k =3.

Original language | English |
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Pages (from-to) | 635-664 |

Journal | Monatshefte für Mathematik |

Volume | 187 |

Issue number | 4 |

Early online date | 17 Jan 2018 |

DOIs | |

Publication status | Published - 1 Dec 2018 |

## Keywords

- Diophantine equations
- Pillai's problem
- Generalized Fibonacci sequence
- Reduction method

## ASJC Scopus subject areas

- Algebra and Number Theory

## Fields of Expertise

- Information, Communication & Computing