Craig William, van Ittersum Jan-Willem, Ono Ken
Department of Mathematics, United States Naval Academy, Annapolis, MD 21402.
Department of Mathematics, University of Virginia, Charlottesville, VA 22904.
Proc Natl Acad Sci U S A. 2024 Sep 24;121(39):e2409417121. doi: 10.1073/pnas.2409417121. Epub 2024 Sep 20.
We show that integer partitions, the fundamental building blocks in additive number theory, detect prime numbers in an unexpected way. Answering a question of Schneider, we show that the primes are the solutions to special equations in partition functions. For example, an integer ≥ 2 is prime if and only if [Formula: see text]where the [Formula: see text] are MacMahon's well-studied partition functions. More generally, for MacMahonesque partition functions [Formula: see text] we prove that there are infinitely many such prime detecting equations with constant coefficients, such as [Formula: see text].
我们证明，整数分拆作为加法数论中的基本组成部分，以一种意想不到的方式检测质数。回答施耐德的一个问题，我们证明质数是分拆函数中特殊方程的解。例如，一个整数(n\geq2)是质数当且仅当(\cdots)（此处原文公式未给出具体内容），其中(\cdots)（此处原文公式未给出具体内容）是麦克马洪研究得很深入的分拆函数。更一般地，对于麦克马洪型分拆函数(\cdots)（此处原文公式未给出具体内容），我们证明存在无穷多个具有常数系数的此类质数检测方程，比如(\cdots)（此处原文公式未给出具体内容）。
