Polynomially growing integer sequences all whose terms are composite

TL;DR

This paper investigates when sequences of the form $a_n = \left\lfloor \dfrac{n^t}{d} \right\rfloor$ are eventually composite, i.e., contain only finitely many primes. It develops a suite of modular and factoring techniques to show primality-free behavior for numerous pairs $(t,d)$, including complete results for several low-degree cases and a universal modulus $d=24$ that works for all even $t\ge4$. The authors also present three general infinite families of prime-free sequences based on Fermat’s little theorem and special forms of primes, and they provide a general Wilson-type argument yielding composite values under certain conditions. Additionally, a substantial table of primitive pairs up to $t=54$ and two illustrative proofs (including a detailed $t=30, d=1116$ case) highlight the ongoing challenge of fully classifying all pairs. The work advances the understanding of how floor-polynomial sequences interact with primality, offering concrete constructions of prime-free sequences and guiding future exploration in this area.

Abstract

We identify pairs of positive integers $(t, d)$ with the property that the integer sequence with general term $\lfloor{n^t/d\rfloor}$ contains at most finitely many primes.

Theorems & Definitions (30)

• proof
• Conjecture
• Theorem 2.1
• proof
• Theorem 3.1
• proof
• Remark
• Lemma 3.1
• proof
• Conjecture