On products of consecutive arithmetic progressions

In this paper, first, we show the Diophantine equationx(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, we raise some unsolved questions.