The number of atoms in an atomic domain

2015/12/16 by Pete L. Clark, Clark, Pete L., Saurabh Gosavi +3
Mathematics · #Analytic Number Theory Research #Advanced Mathematical Identities #Mathematical functions and polynomials

paper · pdf · doi:10.48550/arxiv.1512.05024

Abstract

We study the number of atoms and maximal ideals in an atomic domain with finitely many atoms and no prime elements. We show in particular that for all m,n ∈ ℤ⁺ with n ≥ 3 and 4 ≤ m ≤ (n)/(3) there is an atomic domain with precisely n atoms, precisely m maximal ideals and no prime elements. The proofs use both commutative algebra and additive number theory.

Citations

• Empirical verification of the even Goldbach conjecture and computation of prime gaps up to 4⋅10¹⁸

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