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arXiv · 2607.09105

Extension of the Equation $\sum\limits_{j=1}^{k}jF_{j}^{p}=F_{n}^{q}$ to a Family of Lucas Sequences

Benjamin Earp-Lynch·Simon Earp-Lynch·Omar Kihel·Puntani Pongsumpun
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Abstract

We solve the equation $\sum\limits_{j=1}^{k}jU_{j}(x,y)^{p}=U_{n}(x,y)^{q}$ positive integers $x,p,q,k,n$, with $y=\pm1$ and $\max\{p,q\}\leq11$, where $U_{m}(x,y)=\frac{\alpha^{m}-\beta^{m}}{\alpha-\beta}$ for $\alpha$ and $\beta$ roots of the polynomial $t^2-xt+y$. This generalizes existing results on similar equations, wherein the sequence was fixed as either the Fibonacci or Pell numbers. In addition, we find all solutions with $k=2$ and $y=\pm1$.

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Benjamin Earp-Lynch, Simon Earp-Lynch, Omar Kihel, Puntani Pongsumpun. 2026-07-10. Extension of the Equation $\sum\limits_{j=1}^{k}jF_{j}^{p}=F_{n}^{q}$ to a Family of Lucas Sequences. https://doi.org/10.1080/00150517.2024.12459540

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