I feel like this is solvable using binary to assign which bottles to which prisoners, but only if there were 16 bottles. For example, bottle 11 dec = 1011 bin. So bottle 11 would be tasted by prisoners 1, 3, and 4. Then if 1, 3, and 4 die, we know bottle 11 is poisoned. Every combination of prisoner deaths would point to a unique bottle.
Except 17. Fencepost error on my part maybe?