关于丢番图方程p^x+p^y=z^n

On the Diophantine Equationp^x+p^y=z^n

指数不定方程是一类重要的不定方程,Jesmanowicz-Terai猜想就属于指数不定方程的内容,对此国内外许多学者都对此进行了研究,取得了很多重要的结果。本文利用初等方法,主要是因式分解和简单同余方法并结合Catalan's猜想的基本结论,即两个连续的方幂数有且仅有8和9,得到了一类特殊的指数型丢番图方程p^x+p^y=z^n(n>1)的全部非负整数解,我们证明了该方程当p为奇素数时,它的解与麦什涅素数对应。本文得到的结果改进了Tatong-Suvarnamaniv和Dibyajyoti的结果。

The exponential Diophantine equation is a class of important Diophantine equation,the content of which includes Jesmanowicz-Terai conjecture.Many scholars at home and abroad have studied it and obtained many significant results.In this paper,all non-negative integer solutions of a special exponential Diophantine equation p^x+p^y=z^n(n >1) are obtained by elementary methods (mainly factorization and simple congruence methods) and the basic conclusion of Catalan's conjecture (i.e.there are only 8 and 9 consecutive power numbers).It is proved that its solutions correspond to Mersenne's prime number when p is an odd prime number.The results obtained in this paper have improved the results of Tatong-Suvarnamaniv and Dibyajyoti.