A Note On the Exponential Diophantine Equation (a^n-1)(b^n-1)=x^2

Refik Keskin

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arXiv · 1801.04717

A Note On the Exponential Diophantine Equation (a^n-1)(b^n-1)=x^2

Refik Keskin

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Abstract

In 2002, F. Luca and G. Walsh solved the Diophantine equation in the title for all pairs (a,b) such that 1 4 with 2|n. Also, we solved (a^n-1)(b^n-1)=x^2 for the pairs (a,b)=(2,50),(4,49),(12,45),(13,76),(20,77),(28,49), and (45,100). Lastly, we show that when b is even, the equation (a^n-1)(b^(2n)a^n-1)=x^2 has no solutions n,x.

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Refik Keskin. 2018-01-15. A Note On the Exponential Diophantine Equation (a^n-1)(b^n-1)=x^2. https://arxiv.org/abs/1801.04717

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