| 关于丢番图方程(an)x+(bn)y=(cn)z |
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| 引用本文： | 孙翠芳,汤 敏. 关于丢番图方程(an)x+(bn)y=(cn)z[J]. 数学年刊A辑(中文版), 2018, 39(1): 87-94 |
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| 作者姓名： | 孙翠芳 汤 敏 |
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| 作者单位： | 安徽师范大学数学计算机科学学院 |
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| 摘 要： | 设n,a,b,c是正整数,gcd(a,b,c)=1,a,b≥3,且丢番图方程a~x+b~y=c~z只有正整数解(x,y,z)=(1,1,1).证明了若(x,y,z)是丢番图方程(an)~x+(bn)~y=(cn)~z的正整数解且(x,y,z)≠(1,1,1),则yzz或xzy.还证明了当(a,b,c)=(3,5,8),(5,8,13),(8,13,21),(13,21,34)时,丢番图方程(an)~x+(bn)~y=(cn)~z只有正整数解(x,y,z)=(1,1,1).
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| 关 键 词： | Je''{s}manowicz猜想 丢番图方程 Fibonacci序列 |
| 收稿时间： | 2015-04-06 |
| 修稿时间： | 2016-05-01 |
On the Diophantine Equation (an)x |
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| SUN Cuifa and TANG M. On the Diophantine Equation (an)x[J]. Chinese Annals of Mathematics, 2018, 39(1): 87-94 |
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| Authors: | SUN Cuifa and TANG M |
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| Affiliation: | School of Mathematics and Computer Science, Anhui NormalUniversity, Wuhu 241003, Anhui, China. and School of Mathematics and Computer Science, Anhui NormalUniversity, Wuhu 241003, Anhui, China. |
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| Abstract: | Let $n,a,b,c$ be positive integers with $gcd(a,b,c)=1, a,bgeqslant 3$and the Diophantine equation $a^{x}+b^{y}=c^{z}$ has only the positiveinteger solution $(x,y,z)=(1,1,1)$. In this paper, the authors prove that if$(x,y,z)$ is a positive integer solution of the Diophantine equation$(an)^{x}+(bn)^{y}=(cn)^{z}$ with $(x,y,z)neq (1,1,1)$, then $y
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| Keywords: | Je''{s}manowicz'' conjecture Diophantine equation Fibonacci sequence |
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