Periodic Sequences modulo $m$

Abstract

We give a few remarks on the periodic sequence $a_n=\left(\genfrac{}{}{0ex}{}{n}{x}\right)(modm)$ where $x,m,n\in \mathbb{N}$, which is periodic with minimal length of the period being $$\ell (m,x)=\prod _{i=1}^w{p}_i^{\lfloor {\log }_{p_i}x\rfloor +b_i}=m\prod _{i=1}^w{p}_i^{\lfloor {\log }_{p_i}x\rfloor }$$ where $m={\prod }_{i=1}^w{p}_i^{b_i}$. We prove certain interesting properties of $\ell (m,x)$ and derive a few other results and congruences.

7 pages, preprint. Comments are welcome

References in corpus (1)

• A Simple Number Theoretic Result