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Şiar, Z.; Keskin, R.

On the exponential Diophantine equation \((a^n-2)(b^n-2)=x^2\). (English) Zbl 1496.11058

Math. Notes 111, No. 6, 903-912 (2022).
This paper deals with the solutions of the \((a^n-2)(b^n-2)=x^2\). Most of the existing results find the solutions for certain given values of \((a,b)\). The current authors add to this list by finding all the solutions in some more such cases. They also prove that if \(n\) is even and \(a, b\) are co-prime, then there are no solutions. The authors use the Pell and the first and second kind of Lucas equations to prove their results. They use several classical results on the solutions of the Pell (or Pell like) equations in terms of . In the end they leave the reader with two interesting conjectures, one of which states that if the equation in question has a solution \((n, x)\), then \(n\le 6\)
Reviewer: Anitha Srinivasan (Madrid)

MSC:

11D61 Exponential Diophantine equations

Keywords:

Pell equation; exponential Diophantine equation; Lucas sequence
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References:

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