In this paper, first, we show the Diophantine equation. x(x+b)y(y+b)=z(z+b) has infinitely many nontrivial positive integer solutions for b≥. 3. Second, we prove the Diophantine equation. (x-b)x(x+b)(y-b)y(y+b)=(z-b)z(z+b) has infinitely many nontrivial positive integer solutions for b=. 1, and the set of rational solutions of it is dense in the set of real solutions for b≥. 1. Third, we get infinitely many nontrivial positive integer solutions of the Diophantine equation. (x-b)x(x+b)(y-b)y(y+b)=z2 for even number b≥. 2. At last...
