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The Lewin-Witek conjecture, concerning the index of primitivity of an $n\times n$ matrix, was proved to be false for $n=11$ but to hold for all other $n$. The definitions (which are quite elementary) and history can be found in the answer I posted to https://math.stackexchange.com/questions/450090/if-p-is-a-regular-transition-probability-matrix-then-pn2-has-no-zero-ele