# Two conjectures on the arithmetic in R and C

*Mathematical Logic Quarterly*56 (2):175-189 (2010)

**Abstract**

Let G be an additive subgroup of ℂ, let Wn = {xi = 1, xi + xj = xk: i, j, k ∈ {1, …, n }}, and define En = {xi = 1, xi + xj = xk, xi · xj = xk: i, j, k ∈ {1, …, n }}. We discuss two conjectures. If a system S ⊆ En is consistent over ℝ , then S has a real solution which consists of numbers whose absolute values belong to [0, 22n –2]. If a system S ⊆ Wn is consistent over G, then S has a solution ∈ n in which |xj| ≤ 2n –1 for each j.

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Diophantine Representation of the Set of Prime Numbers.[author unknown]

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2013-12-01

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