On the Diophantine equation $$\sum_{j=1}^{k}jP_j^p=P_n^q$$

Assume that p, q, and k are integers for which the conditions $$1\le p,q\le 10$$ 1 ≤ p , q ≤ 10 and $$2\le k\le 10$$ 2 ≤ k ≤ 10 are satisfied. The initial values $$G_0=0$$ G 0 = 0 , $$G_1=1$$ G 1 = 1 , together with the recursive rule $$G_m=kG_{m-1}+G_{m-2}$$ G m = k G m - 1 + G m - 2 define the non-negative integer sequence $$\{G_m\}_{m=0}^\infty$$ { G m } m = 0 ∞ . In this paper, we solve completely the diophantine equation $$\begin{aligned} G_1^p+2G_2^p+\cdots +\ell G_\ell ^p=G_n^q \end{aligned}$$ G 1 p + 2 G 2 p + ⋯ + ℓ G ℓ p = G n q in the positive integers $$k,p,q,\ell ,n$$ k , p , q , ℓ , n unconditionally for $$\ell$$ ℓ and n. The method works, at least in theory for arbitrary positive integers p, q, and k.

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On the Diophantine equation $$\sum_{j=1}^{k}jP_j^p=P_n^q$$

Cited by 4 publications

References 2 publications

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