solution for equation For $a^2+b^2=c^2$ such that $a, b, c \in \mathbb{Z}$
Do we know whether the solution is finite or infinite for $a, b, c \in \mathbb{Z}$?
We know $a=3, b=4, c=5$ is one of the solutions.

To comment on whether the solution is finite or infinite note that if $(a,b,c)$ satisfy $a^2 + b^2 = c^2$, then so does $(ka,kb,kc)$ where $k \in \mathbb{Z}$. Hence, either no integer solution exists or infinite integer solution exists. 
You have already observed that $(3,4,5)$ satisfies $3^2 + 4^2 = 5^2$. Hence, infinite solutions exist.
Babak Sorouh has given the parameterization, which generates almost all possible solutions $a,b,c \in \mathbb{Z}^+$ (without scaling).
