Periodic Sequences modulo $m$

TL;DR

This paper investigates the periodicity of binomial coefficient sequences modulo m, deriving formulas for their minimal period length and exploring related properties and congruences.

Contribution

It introduces a formula for the minimal period length of binomial coefficient sequences modulo m and analyzes their properties and congruences.

Findings

Derived a formula for the minimal period length of the sequence

Proved properties of the period length and related congruences

Explored implications for binomial coefficient sequences modulo m

Abstract

We give a few remarks on the periodic sequence $a_n=\binom{n}{x}~(mod~m)$ where $x,m,n\in \mathbb{N}$, which is periodic with minimal length of the period being $$\ell(m,x)={\displaystyle\prod^w_{i=1}p^{\lfloor\log_{p_i}x\rfloor+b_i}_i}=m{\displaystyle\prod^w_{i=1}p^{\lfloor\log_{p_i}x\rfloor}_i}$$ where $m=\prod^w_{i=1}p^{b_i}_i$. We prove certain interesting properties of $\ell(m,x)$ and derive a few other results and congruences.