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$$k-$$ k - Fibonacci powers as sums of powers of some fixed primes
Monatshefte für Mathematik ( IF 0.8 ) Pub Date : 2021-02-13 , DOI: 10.1007/s00605-021-01536-6
Carlos A. Gómez , Jhonny C. Gómez , Florian Luca

Let \(S=\{p_{1},\ldots ,p_{t}\}\) be a fixed finite set of prime numbers listed in increasing order. In this paper, we prove that the Diophantine equation \((F_n^{(k)})^s=p_{1}^{a_{1}}+\cdots +p_{t}^{a_{t}}\), in integer unknowns \(n\ge 1\), \(s\ge 1,~k\ge 2\) and \(a_i\ge 0\) for \(i=1,\ldots ,t\) such that \(\max \left\{ a_{i}: 1\le i\le t\right\} =a_t\) has only finitely many effectively computable solutions. Here, \(F_n^{(k)}\) is the nth k–generalized Fibonacci number. We compute all these solutions when \(S=\{2,3,5\}\). This paper extends the main results of [15] where the particular case \(k=2\) was treated.



中文翻译:

$$ k-$$ k-斐波纳契数作为一些固定素数的和

\(S = \ {p_ {1},\ ldots,p_ {t} \} \)为固定的有限质数集,以递增顺序列出。在本文中,我们证明了Diophantine方程\((F_n ^ {(k)})^ s = p_ {1} ^ {a_ {1}} + \ cdots + p_ {t} ^ {a_ {t}} \),以整数未知数\(n \ ge 1 \)\(s \ ge 1,〜k \ ge 2 \)\(a_i \ ge 0 \)表示\(i = 1,\ ldots,t \ ),使得\(\ max \ left \ {a_ {i}:1 \ le i \ le t \ right \} = a_t \)仅具有有限的许多有效可计算解。这里,\(F_N ^ {(K)} \)Ñķ -generalized斐波那契数。当\(S = \ {2,3,5 \} \)时,我们计算所有这些解决方案。本文扩展了[15]的主要结果,其中处理了特定情况\(k = 2 \)

更新日期:2021-02-15
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