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Prove succession

Source: Cono Sur 1993-problem 5

May 30, 2006
inductionmodular arithmeticnumber theory unsolvednumber theory

Problem Statement

Prove that there exists a succession a1,a2,...,ak,...a_1, a_2, ... , a_k, ..., where each aia_i is a digit (ai∈(0,1,2,3,4,5,6,7,8,9)a_i \in (0, 1, 2, 3, 4, 5, 6, 7, 8, 9) ) and a0=6a_0=6, such that, for each positive integrer nn, the number xn=a0+10a1+100a2+...+10n−1an−1x_n=a_0+10a_1+100a_2+...+10^{n-1}a_{n-1} verify that xn2−xnx_n^2-x_n is divisible by 10n10^n.
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