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arXiv · 2405.12310

Sparse Admissible Sets and a Problem of Erd\H{o}s and Graham

Desmond Weisenberg
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Abstract

Erd\H{o}s and Graham asked whether any sparse enough admissible set of natural numbers can be translated into a subset of the primes. By using a greedy construction involving powers of primitive roots, we prove that there exist arbitrarily sparse infinite admissible sets that cannot be translated into a subset of the primes, thus answering this question in the negative. We present three additional constructions as well.

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Desmond Weisenberg. 2024-05-20. Sparse Admissible Sets and a Problem of Erd\H{o}s and Graham. https://doi.org/10.5281/zenodo.13909172

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