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Proving that a fraction is a national number

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January 3, 2011
number theory unsolvednumber theory

Problem Statement

For a positive integer nn, let 6(n)6^{(n)} be the natural number whose decimal representation consists of nn digits 66. Let us define, for all natural numbers mm, kk with 1≤k≤m1 \leq k \leq m [mk]=6(m)6(m−1)⋯6(m−k+1)6(1)6(2)⋯6(k).\left[\begin{array}{ccc}m\\ k\end{array}\right] =\frac{ 6^{(m)} 6^{(m-1)}\cdots 6^{(m-k+1)}}{6^{(1)} 6^{(2)}\cdots 6^{(k)}} . Prove that for all m,km, k, [mk] \left[\begin{array}{ccc}m\\ k\end{array}\right] is a natural number whose decimal representation consists of exactly k(m+k−1)−1k(m + k - 1) - 1 digits.
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