Strong arithmetic property of certain Stern polynomials

TL;DR

This paper explores the properties of Stern polynomials, focusing on the existence of odd solutions to specific polynomial congruences, and demonstrates infinite solutions for certain modular conditions.

Contribution

It provides new results on the existence of infinitely many odd solutions to particular congruences involving Stern polynomials for specific moduli and parameters.

Findings

Infinitely many odd solutions for m=2, r=0 or 1

Infinite solutions for m=3, r=0

Numerical evidence supporting theoretical results

Abstract

Let $B_{n}(t)$ be the $n$th Stern polynomial, i.e., the $n$th term of the sequence defined recursively as $B_{0}(t)=0, B_{1}(t)=1$ and $B_{2n}(t)=tB_{n}(t), B_{2n+1}(t)=B_{n}(t)+B_{n-1}(t)$ for $n\in\N$. It is well know that $i$th coefficient in the polynomial $B_{n}(t)$ counts the number of hyperbinary representations of $n-1$ containing exactly $i$ digits 1. In this note we investigate the existence of odd solutions of the congruence \begin{equation*} B_{n}(t)\equiv 1+rt\frac{t^{e(n)}-1}{t-1}\pmod{m}, \end{equation*} where $m\in\N_{\geq 2}$ and $r\in\{0,\ldots,m-1\}$ are fixed and $e(n)=\op{deg}B_{n}(t)$. We prove that for $m=2$ and $r\in\{0,1\}$ and for $m=3$ and $r=0$, there are infinitely many odd numbers $n$ satisfying the above congruence. We also present results of some numerical computations.