Remainders Are Fun 5

$\begin{aligned} S & = \sum_{i=1}^{2017} \sum_{p=1}^i \sqrt{157^2+32(8p+3)(p+20)} \\ & = \sum_{i=1}^{2017} \sum_{p=1}^i \sqrt{24649 +32 (8p^2+163p+60)} \\ & = \sum_{i=1}^{2017} \sum_{p=1}^i \sqrt{256p^2+5216p+26569} \\ & = \sum_{i=1}^{2017} \sum_{p=1}^i \sqrt{\left(16p+163\right)^2} \\ & = \sum_{i=1}^{2017} \sum_{p=1}^i \left(16p+163\right) \\ & = \sum_{i=1}^{2017} \left(8i(i+1)+163i\right) \\ & = \sum_{i=1}^{2017} \left(8i^2+171i\right) \\ & = \frac {8(2017)(2018)(4035)}6 + \frac {171(2017)(2018)}2 \\ & = 8(2017)(1009)(1345) + 171(2017)(1009) \\ & = (2017)(1009)(10760+171) \\ & = (2017)(1009)(10931) \\ & = (17)(9)(931) \text{ (mod 1000)} \\ & = (17)(9)(-69) \text{ (mod 1000)} \\ & = -557 \text{ (mod 1000)} \\ & = \boxed{443} \text{ (mod 1000)} \end{aligned}$