本原商高数的Jesmanowicz猜想的位移形式  

The Shuffle Variant of Jesmanowicz Conjecture

作　　者：冯强[1] 

机构地区：[1]延安大学数学与计算机学院,陕西延安716000

出　　处：《数学的实践与认识》2015年第16期312-315,共4页Mathematics in Practice and Theory

基　　金：陕西省高水平大学建设基金资助项目(2012SXTS07);陕西省教育厅基金项目(12jk0893)

摘　　要：一组正整数(a,b,c)称为本原商高数,如果它们满足方程a^2+b^2=c^2且(a,b)=1,2|b.著名的Jesmanowicz-Terai猜想是指当(a,b,c)是本原商高数时,方程a^x+b^y=c^z仅有正整数解(x,y,z)=(2,2,2).本文讨论了商高数的位移形式,即就是:设u是大于2的偶数,本文运用初等数论方法以及同余的性质讨论了指数Diophantine方程(u^2+1)~x+(2u)~y=(u^2-1)~z的可解性,证明了该方程无正整数解(x,y,z).从而部分的解决了Jesmanowicz-Terai猜想的另一种形式.It is well known that（a,b,c） is called the primitive Pythagorean numbers,if a^2 ＋ b^2 = c^2 with（a,b） = 1 and 2｜6.The famous Jesmanowicz-Terai conjecture is that for any primitive Pythagorean numbers（a,b,c）,the diophantine equation a^x ＋b^y = c^z has only positive integer solution（x,y,x） —（2,2,2）.In paper,we considered the shuffle variant of JeSmanowicz-Terai conjecture.That is,let u be an even integer with u〉 2,this paper,we using some elementary number theory methods,the properties of congruence and the exponential diophantine equation to study the solvability of the exponential diophantine equation（u^2 ＋1）~x ＋（2u）~y =（u^2 —1）~z,and prove that the equation has no positive integer solutions（x,y,z）.This partly solved the another form of Jesmanowicz-Terai conjecture.

关 键 词：指数DIOPHANTINE方程 正整数解 Jesmanowicz-Terai猜想 位移形式 

分 类 号：O156[理学—数学]