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The complete solution of the Diophantine equation \((4m^2+1)^x+(5m^2-1)^y=(3m)^z\)

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Abstract

In this short note we give all solutions to the exponential Diophantine equation \((4m^2+1)^x+(5m^2-1)^y=(3m)^z\) where \(20<m<90\) and \(m\equiv 3\) \((mod\quad 6)\). In view of earlier theorems of Terai, and of Su and Li, this result yields a complete solution set to a problem of Terai.

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Acknowledgments

The author is grateful to the referee for her/his useful remark, and to Attila Pethő for calling his attention to the problem. Research supported by the TÁMOP-4.2.4.A/2-11-1-2012-0001 project. The project has been supported by the European Union, co-financed by the European Social Fund.

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  1. Institute of Mathematics, University of Debrecen, P.O. Box 12, Debrecen, 4010, Hungary

    Csanád Bertók

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  1. Csanád Bertók
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Bertók, C. The complete solution of the Diophantine equation \((4m^2+1)^x+(5m^2-1)^y=(3m)^z\) . Period Math Hung 72, 37–42 (2016). https://doi.org/10.1007/s10998-016-0111-x

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