On the exponential Diophantine equation $$(3pm^2-1)^x+(p(p-3)m^2+1)^y=(pm)^z$$ |
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Authors: | Nobuhiro Terai Takeshi Hibino |
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Institution: | 1.Department of Computer Science and Intelligent Systems, Faculty of Engineering,Oita University,Oita,Japan;2.Department of Psychology and Child Studies, Faculty of Human Sciences,University of East Asia,Shimonoseki,Japan |
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Abstract: | Let m be a positive integer, and let p be a prime with \(p \equiv 1~(\mathrm{mod}~4).\) Then we show that the exponential Diophantine equation \((3pm^2-1)^x+(p(p-3)m^2+1)^y=(pm)^z\) has only the positive integer solution \((x, y, z)=(1, 1, 2)\) under some conditions. As a corollary, we derive that the exponential Diophantine equation \((15m^2-1)^x+(10m^2+1)^y=(5m)^z\) has only the positive integer solution \((x, y, z)=(1, 1, 2).\) The proof is based on elementary methods and Baker’s method. |
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