= Solution
It suffices to take four players. Both two-player <coalitions> $\{1,2\}$ and $\{3,4\}$ win, because two is a <prime number>, while the empty <coalition> and the four-player <coalition> lose.
Suppose real weights and quota $q$ represented this game. The empty <coalition> loses, so $0<q$. The two winning <coalitions> require
$$
w_1+w_2\ge q,\qquad w_3+w_4\ge q.
$$
Adding gives
$$
w_1+w_2+w_3+w_4\ge2q>q,
$$
so the four-player <coalition> must win, a contradiction. Hence \b[prime-cardinality games cannot in general be represented as <weighted voting games>]. The same four-player obstruction appears inside every such game with at least four players.
This proof permits negative individual weights, as the printed definition does. An argument based solely on monotonicity would require a nonnegative-weight assumption and would not suffice for that definition.
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