关于丢番图方程P^(2z)-P^zD^m+D^(2m)=X^2(I)

On the Diophantine Equation P^(2z)-P^zD^m+D^(2m)=X^2(I)

设P为素数,P鄹D>1,完全解决丢番图方程A:P2z-PzDm+D2=X2。得到如下结论:(Ⅰ)若P=2,则方程(A)除D=3仅有非负整数解26-23·3+32=72和D=3·22k-4+2k-1-1(k≥3)仅有非负整数解22k-2k·(3·22k-4+2k-1-1)+(3·22k-4+2k-1-1)2=(3·22k-4+1)2以及D=22k-4+2k-1-3(k≥3)仅有非负整数解22k-2k·(22k-4+2k-1-3)+(22k-4+2k-1-3)2=(22k-4+3)2之外,无其他非负整数解。(Ⅱ)若P=3,则方程(A)除D=32k+1+2·3k-14(k≥1)仅有非负整数解32k-3k·32k+1+2·3k-14+(32k+1+2·3k-14)2=32k+1+14 2之外,无其他非负整数解。(Ⅲ)若P>3为奇素数熏则方程(A)除D=3P2k+2Pk-34(k≥1)仅有非负整数解P2k-Pk·P2k+2Pk-34+(P2k+2Pk-34)2=3P2k+14 2和D=P2k+2Pk-34(k≥1)仅有非负整数解P2k-Pk·P2k+2Pk-3+(P2k+2Pk-3)2=P2k+3 2之外,无其他非负整数解。

Let P is a odd prime,P D>1.We have already sovled the diophantine equation(A):P2z-PzDm+D2=X2.We have:(Ⅰ)If P=2, then equation(A)has not other nonnegative integral solution except when D=3 equation (A) only has nonnegative integral solution 26-23·3+32=72and when D=3·22k-4+2k-1-1(k≥3) equation (A) only has nonnegative integral solution 22k-2k·(3·22k-4+2k-1-1)+(3·22k-4+2k-1-1)2=(3·22k-4+1)2 and when D=22k-4+2k-1-3(k≥3) equation (A) only has nonnegative integral solution 22k-2k·(22k-4+2k-1-3)+(22k-4+2k-1-3)2=(22k-4+3)2.(Ⅱ)If P=3,then equation(A)has not other nonnegative integral solution except when D=(k≥1) equation (A) only has nonnegative integral solution 32k-3k·++()2= .(Ⅲ)If P>3,P is a odd prime, then equation(A)has not other nonnegative integral solution.except when D=(k≥1)equation only has nonnegative integral solution P2k-Pk·++( ) = and when D= (k≥1)equation only has nonnegative integral solution P2k-Pk· +( )2= .