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Lucas

February 20, 2013 Number Theory Number Theory

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$p$ $ab\in\textrm{N}_+$

$a=a_k p^k+a_{k-1}p^{k-1}+\cdots+a_1 p+a_0$

$b=b_k p^k+b_{k-1}p^{k-1}+\cdots+b_1 p+b_0$

$0\leqslant a_i,b_i\leqslant p-1$ $i=0,1,2,\cdots,k$

$C_a^b=C_{a_k}^{b_k}\cdot C_{a_{k-1}}^{b_{k-1}}\cdot \cdots \cdot C_{a_0}^{b_0}\left(\bmod p\right)$

$i\in\{0,1,2,\cdots,k\}$ $b_i>a_i$ $C_a^b\equiv 0(\bmod p)$

$C_a^b$ $a$ $b$

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