关于商高数的Je\'{s}manowicz猜想

On Je\'{s}manowicz' Conjecture Concerning Pythagorean Triples

中文关键词:  Je'smanowicz猜想  丢番图方程

英文关键词:Je\'{s}manowicz' conjecture  Diophantine equation

中文摘要:

      设$(a,b,c)$是本原商高数组, 1956年Je'smanowicz猜想对任意正整数$n$, 丢番图方程$(an)^x+(bn)^y=(cn)^z$仅有正整数解$(x,y,z)=(2,2,2)$. 设素数$p\equiv 3\pmod 4$, $s$ 是正整数. 本文中我们证明了当$(a,b,c)=(4p^{2s}-1,4p^{s},4p^{2s}+1)$并且满足某些整除条件时, 该猜想是正确的.

英文摘要:

      Let $(a,b,c)$ be a primitive Pythagorean triple. Je\'{s}manowicz conjectured in 1956 that for any positive integer $n$, the Diophantine equation $(an)^x+(bn)^y=(cn)^z$ has only the positive integer solution $(x,y,z)=(2,2,2)$. Let $p\equiv 3\pmod 4$ be a prime and $s$ be some positive integer. In the paper, we show that the conjecture is true when $(a,b,c)=(4p^{2s}-1,4p^{s},4p^{2s}+1)$ and certain divisibility conditions are satisfied.