# On the period mod $m$ of polynomially-recursive sequences: a case study

## TL;DR

This paper investigates the periodic behavior modulo m of second order polynomially-recursive sequences, linking number theory concepts and providing generalized supercongruences for these sequences.

## Contribution

It introduces new results on the period mod m of polynomially-recursive sequences and extends supercongruences to a broader class of recurrences.

## Key findings

- Analysis of period mod m for second order sequences
- Connection with number theory concepts like Carmichael function and Wieferich primes
- Generalization of supercongruences to broader recurrence classes

## Abstract

Polynomially-recursive sequences generally have a periodic behavior mod $m$. In this paper, we analyze the period mod $m$ of a second order polynomially-recursive sequence. The problem originally comes from an enumeration of avoiding pattern permutations and appears to be linked with nice number theory notions (the Carmichael function, Wieferich primes, algebraic integers). We give the mod $a^k$ supercongruences, and generalize these results to a class of recurrences.