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Decision over the irrationality of the roots of the simple indicator with forms as $a^x+b^x=c^x$

MinXue Niu No.2 Secondary School attached to East China Normal University RuoQing Cai No.2 Secondary School attached to East China Normal University

S.-T. Yau High School Science Awarded Papers mathscidoc:1608.35148

Dongrun-Yau Science Award, 2013
Starting from the roots of the equation 3^x + 4^x = 5^x discussed in the Tenth Grade, we utilized the method, the algebraic extension of the rational number eld, to produce the ways to judge whether the roots of the basic exponential equation with a form as a^x+b^x = c^x are rational or not. For equations with more than two terms on the left side, as is the equation a_1^x+a_2^x+ยทยทยท +a_n^x = d^x, the determination of whether the root was irrational was comparatively dicult.Therefore, we provided a prevalent method for the examination of the root of a three-term equation as well as a conclusion that if the equation doesn't have integer roots, the roots won't be a rational number with a denominator of two. Finally, based on the method, the algebraic extension of the rational number eld, we concluded that under special occasions, the root of the equation can't be some rational numbers with certain denominators.
Keywords:exponential equation, algebraic extension, unit root
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@inproceedings{minxue2013decision,
  title={Decision over the irrationality of the roots of the simple indicator with forms as $a^x+b^x=c^x$},
  author={MinXue Niu, and RuoQing Cai},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160818151441104777242},
  booktitle={Dongrun-Yau Science Award},
  year={2013},
}
MinXue Niu, and RuoQing Cai. Decision over the irrationality of the roots of the simple indicator with forms as $a^x+b^x=c^x$. 2013. In Dongrun-Yau Science Award. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160818151441104777242.
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