I think nobody pointed this problem, if it is repeated, please say me to delete it. This problem killed me for three weeks, when I was a young student in high school. So, I want to recall it again.
$Problem:$ Find all right triangles with rational sides, where the area of these triangles are integer?
I think it is still open problem and if somebody can solve it, I will give 100$ as a small award.
After I searched, I found these two interesting sources. I hope it will be helpful.
1) N.Koblitz, Introduction to elliptic curves and modular forms, volume 97 of Graduate Texts in Mathematics. Springer-Verlag, New York, second edition, 1993.
2) Washington, Lawrence C., Elliptic Curves : Number Theory and Cryptography, CRC Press Series On Discrete Mathematics and Its Applications