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Sumsets with restricted number of prime factors

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Abstract

In 1934, Erdős and Turán proved that if a1, a2, . . . , an are distinct positive integers such that the product Π1≤i<j≤n(ai + aj) has at most k distinct prime factors, then n < 3 × 2k − 1. In this paper, we introduce the notion of the p-graph of a set of positive integers and improve the bound to n ≤ 2k. In particular, we prove that the bound 2k is optimal for k = 2. Beyond these, we also obtain some related results and pose several problems for further research.

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  1. Aliyun School of Big Data, Changzhou University, Changzhou, 213164, China

    Bing-Ling Wu

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  1. Bing-Ling Wu
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Wu, BL. Sumsets with restricted number of prime factors. Lith Math J 59, 251–260 (2019). https://doi.org/10.1007/s10986-019-09441-0

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  • DOI: https://doi.org/10.1007/s10986-019-09441-0

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