For a canonical threefold $X$, it is known that $p_n$ does not vanish for a sufficiently large $n$, where $p_n=h^0(X,{mathcal O}_X(nK_X))$. We have shown that $p_n$ does not vanish for at least one $n$ in ${6,,8,,10}$. Assuming an additional condition $p_2geq 1$ or $p_3geq 1$, we have shown that $p_{12}geq 2$ and $p_ngeq 2$ for $ngeq 14$ with one possible exceptional case. We have also found some inequalities between $chi({mathcal O}_X)$ and $K_X^3$.
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