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Given a non-negative number n, what is the best approach to find the most compact representation for n in terms of sums of powers, such that the bases and the exponents can't surpass a given value (...

I want to ask about decidability of exponential Diophantine equation: $z_12^{\eta_1} + \ldots + z_n2^{\eta_n} = z$, where $z_i,\eta_i,z \in \mathbb{Z}$, and $\eta_i$ - are variables. Can we find ...