$20172017^2$

Note that $20172017^2=73^2\times 137^2\times 2017^2$ , and hence $20172017^2$ has $(2+1)(2+1)(2+1)=27$ positive factors. Also, $(c-b)(c+b)=20172017^2,$ and so we are looking for two positive factors $(c-b)$ and $(c+b)$ whose product is $20172017^2$ . There are $27$ ways to assign these factors, but $13$ will give $b$ negative, and one will give $b=0$ (when $(c-b)=(c+b)=20172017$ ). This leaves $13$ ways to assign the factors so that $b$ and $c$ are positive, meaning there are $\boxed{13}$ positive, integral solutions for $(b,c)$ .