Remark on Pascal's Triangle

TL;DR

This paper explores the fractal structure of Pascal's triangle modulo a prime p, using Kummer's theorem to connect divisibility with carries in base p addition, through elementary exercises.

Contribution

It provides an elementary explanation of Pascal's triangle's fractal pattern modulo p based on Kummer's theorem and base p arithmetic.

Findings

Reveals the fractal structure of Pascal's triangle modulo p.

Connects divisibility in binomial coefficients with carries in base p addition.

Uses elementary exercises to illustrate the theorem's implications.

Abstract

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 $p$.