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China TST 1986 construction of sequence

Source: China TST 1986, problem 3

May 16, 2005
algebraNumber theoretic functionsChinaTSTdecimal representationDigits

Problem Statement

Given a positive integer AA written in decimal expansion: (an,an−1,…,a0)(a_{n},a_{n-1}, \ldots, a_{0}) and let f(A)f(A) denote ∑k=0n2n−k⋅ak\sum^{n}_{k=0} 2^{n-k}\cdot a_k. Define A1=f(A),A2=f(A1)A_1=f(A), A_2=f(A_1). Prove that:
I. There exists positive integer kk for which Ak+1=AkA_{k+1}=A_k. II. Find such AkA_k for 1986.19^{86}.
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