  We prove that a curve of degree $dk$ on a very general surface of degree $d
\geq 5$ in $\mathbb{P}^3$ has geometric genus at least $\frac{dk(d-5)+k}{2} +
1$. This improves bounds given by G. Xu. As a corollary, we conclude that the
very general quintic surface in $\mathbb{P}^3$ is algebraically hyperbolic.
