A New Polygonal Number Theorem

IF 0.4 4区数学 Q4 MATHEMATICS

American Mathematical Monthly Pub Date : 2023-01-09 DOI:10.1080/00029890.2022.2159256

Benjamin Lee Warren

引用次数: 0

Abstract

The polygonal number theorem of Fermat, Cauchy, and Legendre has served as one of the leading results in the history of additive number theory. It states that every positive integer can be written as the sum of m m-gonal numbers, and Legendre improved this to four or five m-gonal numbers for sufficiently large integers. A variation of this problem is to determine the minimal amount of m-gonal numbers needed in order to represent all integers as the sum and difference of these elements infinitely many different ways. Fortunately, a full solution is provided to this problem as the following result.

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一个新的多边形数定理

费马、柯西和勒让德的多边形数定理是加性数论历史上的主要成果之一。它指出，每个正整数都可以写成m个m形数的和，勒让德将其改进为足够大的整数的4或5个m形数。这个问题的另一种变化是，确定将所有整数表示为这些元素的无穷多种不同方式的和与差所需的最小m形数。幸运的是，这个问题的完整解决方案如下所示。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

American Mathematical Monthly Mathematics-General Mathematics

CiteScore

0.80

自引率

20.00%

发文量

127

审稿时长

6-12 weeks

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