关于Diophantine方程px+(p+2)~y=z^2的一个注记（英文）

A note on the Diophantine equation px+(p+2)~y=z^2

设p是一个固定的奇素数.为了得到方程px+(p+2)y=z2的解,利用指数Diophantine方程和二次剩余的性质证明了当2y时,方程px+(p+2)y=z2无解(x,y,z);此外,如果p≡±3(mod 8)或p≠7,并且p+2是一个平方数,那么,方程也无解(x,y,z).

Let pbe a fixed odd prime.In order to get the solutions of equation px＋（p＋2）y=z2,by using the properties of exponential Diophantine equation and quadratic residue,it is proved that equation px＋（p＋2）y=z2 has no solution（x,y,z）with 2y.In addition,if either p≡±3（mod 8）or p≠7and p＋2is a square,then the equation also has no positive integer solution（x,y,z）.