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Hard base 10 number theory

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December 15, 2019
number theorybase 10AM-GM

Problem Statement

Prove that for any natural number n, n, there exists a number having n+1 n+1 decimal digits, namely, k0,k1,k2,…,kn k_0,k_1,k_2,\ldots ,k_n , and a (n+1)-tuple, \text{(n+1)-tuple}, say (ϵ0,ϵ1,ϵ2…,ϵn)∈{−1,1}n+1,\left( \epsilon_0 ,\epsilon_1 ,\epsilon_2\ldots ,\epsilon_n \right)\in\{-1,1\}^{n+1} , that satisfies: 1≤∏j=0n(2+j)kj⋅ϵj≤210n−1 1\le \prod_{j=0}^n (2+j)^{k_j\cdot \epsilon_j}\le \sqrt[10^n-1]{2}
Sorin Rădulescu and Ion Savu
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