An Arithmetic Characterization of 2-Generated Numbers

Summary · qwen2.5:32b

The article establishes an arithmetic criterion for identifying 2-generated numbers based on their prime factorization, where a number $n$ is a 2-generated number if every group of order $n$ can be generated by two elements. This characterization is significant as it provides a clear condition to determine the generation property of groups solely through the lens of number theory. For instance, all square-free numbers are identified as 2-generated under specific conditions related to their prime factors.

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Excerpt

arXiv:2508.16356v2 Announce Type: replace Abstract: A group $G$ is said to be $k$-generated if it has a generating set with $k$ elements. A positive integer $n$ is called a \emph{2-generated number} if every group of order $n$ is 2-generated. In this article, we establish an arithmetic characterization of 2-generated numbers expressed in terms of the prime factorization of $n$.