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On some ternary pure exponential diophantine equations with three consecutive positive integers bases

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Abstract

By using the lower bound of linear forms in two logarithms of Laurent (Acta Arith. 133(4) (2008) 325–348), we give here a new solution that the ternary pure exponential diophantine equation \((n+1)^{x}+(n+2)^{y}=n^{z}\) has no positive integer solutions except for \((n,x,y,z)=(3,1,1,2)\). This proof is very different from Le (J. Yulin Teachers College 28(3) (2007) 1–2), in which he used the classification method of solutions of exponential decomposition form equation. Furthermore, we solved completely another similar ternary pure exponential diophantine equation \(n^{x}+(n+2)^{y}=(n+1)^{z}\) by using m-adic estimation of linear forms due to Bugeaud (Compos. Math. 132(2) (2002) 137–158).

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Acknowledgements

The authors would like to thank Professor Yuri Bilu for fixing (3.13). The second and fourth authors were supported by China National Nature Foundation (Grant Nos 11301363 and 11501477), The Science Fund of Fujian Province (Grant No. 2015J01024), the Fundamental Research Funds for the Central University (Grant No. 2072017001) and the Sichuan Provincial Scientific Research and Innovation Team in Universities (No. 14TD0040).

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Authors and Affiliations

  1. School of Science, Xi’an Shiyou University, Xi’an, 710065, Shanxi, People’s Republic of China

    Ruiqin Fu

  2. Institute of Mathematics, Aba Teachers University, Wenchuan, 623000, Sichuan, People’s Republic of China

    Bo He

  3. School of Science, Xi’an Polytechnic University, Xi’an, 710048, Shanxi, People’s Republic of China

    Hai Yang

  4. School of Mathematical Sciences, Xiamen University, Xiamen, 361005, Fujian, People’s Republic of China

    Huilin Zhu

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  1. Ruiqin Fu
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  2. Bo He
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  3. Hai Yang
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  4. Huilin Zhu
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Correspondence to Huilin Zhu.

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Communicating Editor: B Sury

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Fu, R., He, B., Yang, H. et al. On some ternary pure exponential diophantine equations with three consecutive positive integers bases . Proc Math Sci 129, 26 (2019). https://doi.org/10.1007/s12044-019-0468-x

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