| 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,2018,39(1):87~94 |
| 关于丢番图方程(an)x+(bn)y=(cn)z |
| On the Diophantine Equation (an)x |
| Received:April 06, 2015 Revised:May 01, 2016 |
| DOI：10.16205/j.cnki.cama.2018.0009 |
| 中文关键词: Je'{s}manowicz猜想, 丢番图方程, Fibonacci序列 |
| 英文关键词:Je'{s}manowicz' conjecture, Diophantine equation, Fibonacci sequence |
| 基金项目: |
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| 中文摘要: |
| 设$n,a,b,c$是正整数, $\gcd(a,b,c)=1,\ a,b\geqslant 3$,
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$(x,y,z)\neq (1,1,1)$, 则$y |
| 英文摘要: |
| Let $n,a,b,c$ be positive integers with $\gcd(a,b,c)=1,\ a,b\geqslant 3$
and the Diophantine equation $a^{x}+b^{y}=c^{z}$ has only the positive
integer 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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