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We construct an algorithm to test if a system of conditions of the types μ<η, μ/η, μ=η a , P a (μ), ¬(μ<η), ¬(μ/η), ¬(μ=η a ), and ¬(P a (μ)) has a solution in natural numbers. (a∈N, and P a denotes the set {n a :n∈N}.)

Decidability of the existential theory of the set of natural numbers with order, divisibility, power functions, power predicates, and constants