Abstract
In this paper, we deal with the equation \((a^{n}-2)(b^{n}-2)=x^{2}\), \(2\leq a<b\), and \(a,b,x,n\in\mathbb{N}\). We solve this equation for \((a,b)\in\{(2,10),(4,100),(10,58),(3,45)\}\). Moreover, we show that \((a^{n}-2)(b^{n}-2)=x^{2}\) has no solution \(n,x\) if \(2|n\) and \(\gcd(a,b)=1\). We also give a conjecture which says that the equation \((2^{n}-2)((2P_{k})^{n}-2)=x^{2}\) has only the solution \((n,x)=(2,Q_{k})\), where \(k>3\) is odd and \(P_{k},Q_{k}\) are the Pell and Pell Lucas numbers, respectively. We also conjecture that if the equation \((a^{n}-2)(b^{n}-2)=x^{2}\) has a solution \(n,x,\) then \(n\leq6\).
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We would like to thank the anonymous referee for very helpful comments and suggestions.
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Şiar, Z., Keskin, R. On the Exponential Diophantine Equation \((a^{n}-2)(b^{n}-2)=x^{2}\). Math Notes 111, 903–912 (2022). https://doi.org/10.1134/S0001434622050248
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DOI: https://doi.org/10.1134/S0001434622050248