Archimedes's cattle problem
Sledgehammer, 67 66 26 bytes
-40 bytes thanks to @GregMartin and I also no longer have any idea about how my answer works
Completes in less than a few seconds!
⡇⣄⠀⠇⣺⠇⢞⡞⣵⣍⠪⢺⡇⠜⢂⡒⢃⠦⠲⣎⠇⠣⡔⢻⡦⠔

Mathematica code:
Floor[Divide[25194541,184119152] * (NumberFieldFundamentalUnits@Sqrt[4729494])^4658]

(the reason for the Divide is that by default Mathematica represents x/y as x * y^-1, which is half a byte longer in Sledgehammer).