On other two representations of the C-recursive integer sequences by terms in modular arithmetic

TL;DR

This paper explores alternative representations of C-recursive integer sequences using modular arithmetic, specifically through compositions of remainder operations and division applied to remainders, expanding on previous work.

Contribution

It introduces new forms of representing C-recursive sequences by combining division and remainder operations, providing novel insights into their structure.

Findings

New representations using division and remainders are proposed.

These forms extend previous modular representations of recursive sequences.

The approach offers potential for simplified analysis of such sequences.

Abstract

An integer sequence that is defined by initial values and a linear recurrence with constant integer coefficients, can be represented by the difference of two arithmetic terms containing exponentiation. All constants occuring in the term are integers. While in the paper "On the representation of C-recursive integer sequences by arithmetic terms" by Prunescu and Sauras-Altuzarra, the terms consist of the remainder operation, applied on a division; the representations shown here are a division applied to a remainder operation, respectively the composition of two remainder operations.