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On the length over which $k$-Göbel sequences remain integers

Yuh Kobayashi, Shin-ichiro Seki

TL;DR

The paper resolves the unboundedness question for the length of integers in $k$-Göbel sequences by proving that $N_k$ is unbounded. It uses an elementary local-global strategy: for each $m≥4$ they select $k$ in a residue class tied to $m$ and the primorial $m ext{#}$, then analyze localized sequences $g_{k_p,p,r_p}(n)$ at primes $p≤m$ to force a non-integer term before or at $m$. They show that for all such $k$, the condition $g_{k_p,p,r_p}(m) ≠ F$ holds (verified for $p=2$ and odd primes), implying $N_k>m$ and consequently $ ext{sup}_k N_k = ∞$. The argument extends to the $(k,l)$-Göbel family, showing the unboundedness result holds under fixed initial values $l$ as well.

Abstract

We prove that the sequence $(N_k)_k$, where each $N_k$ is defined as the smallest positive integer $n$ for which the $n$th term $g_{k,n}$ of the $k$-Göbel sequence is not an integer, is unbounded.

On the length over which $k$-Göbel sequences remain integers

TL;DR

The paper resolves the unboundedness question for the length of integers in -Göbel sequences by proving that is unbounded. It uses an elementary local-global strategy: for each they select in a residue class tied to and the primorial , then analyze localized sequences at primes to force a non-integer term before or at . They show that for all such , the condition holds (verified for and odd primes), implying and consequently . The argument extends to the -Göbel family, showing the unboundedness result holds under fixed initial values as well.

Abstract

We prove that the sequence , where each is defined as the smallest positive integer for which the th term of the -Göbel sequence is not an integer, is unbounded.

Paper Structure

This paper contains 3 sections, 1 theorem.

Table of Contents

  1. Introduction
  2. Proof
  3. Remark

Key Result

Theorem 1

Let $m$ be a positive integer. If $k\geq 2$ satisfies $k\equiv 1 \pmod{m!/m\#}$, then $N_k > m$. In particular, $\sup_{k\geq 2}N_k=\infty$.

Theorems & Definitions (2)

  • Theorem
  • proof : Proof of Theorem