一个丢番图方程的求解

Solution of a diophantine equation

设P为素数,P D>1,证明了丢番图方程P2z-PzDm+D2m=X2除D=2仅有非负整数解32-3·23+26=72和2 D,P≡1(mod8)时,必有m=2,z=1或m=1以及2|D,D含2hq+1形素因子(这里q|m,q为奇素数,h≥1)之外,推出m=1.

Let P is a odd prime,PD>1,We have solved diophantine equation P^(2z)-P^zD^m+D^(2m)=X^2,certainly has m=1 except when D=2 this equation only has nonnegative integral solution 3~2-3·2~3+2~6=7~2 and when 2D,P≡1(mod8) this equation certainly has m=2,z=1 or m=1,and when 2｜D,(2hq+1)｜D,where 2hq+1 and q are odd primes,q｜m.