帕斯卡三角形备注

arXiv - MATH - History and Overview Pub Date : 2024-05-20 DOI:arxiv-2405.13060

Chaim Goodman-Strauss

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引用次数: 0

摘要

通过一系列基本练习，我们利用库默尔在 1852 年提出的一个定理来解释帕斯卡三角形在模数为 $p$ 时的分形结构：当且仅当一个素数 $p$ 在以 $p$ 为基数的加法中携带 $i+j=n$ 时，这个素数 $p$ 除以 $\dfrac {n!}{i!j!｝当且仅当以 $p$ 为基数书写时，加法 $i+j=n$ 中有一个进位时，才会出现 $ddfrac{n!}{i!j!}。

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Remark on Pascal's Triangle

Through a series of elementary exercises, we explain the fractal structure of Pascal's triangle when written modulo $p$ using an 1852 theorem due to Kummer: A prime $p$ divides $\dfrac {n!}{i!j!} $ if and only if there is a carry in the addition $i+j=n$ when written in base $p$.

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arXiv - MATH - History and Overview

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