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arXiv · 1707.07519

On a problem of Pillai with $k$-generalised Fibonacci numbers and powers of $2$

Mahadi Ddamulira·Carlos A. Gómez·Florian Luca
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Abstract

For an integer $ k\geq 2 $, let $ \{F^{(k)}_{n} \}_{n\geq 0}$ be the $ k$--generalized Fibonacci sequence which starts with $ 0, \ldots, 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 powers of 2 for any fixed $k \geqslant 4$. This paper extends previous work from [9] for the case $k=2$ and [6] for the case $k=3$.

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Mahadi Ddamulira, Carlos A. Gómez, Florian Luca. 2017-11-04. On a problem of Pillai with $k$-generalised Fibonacci numbers and powers of $2$. https://doi.org/10.1007/s00605-018-1155-1

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