$(1^3+1)(2^3+1)\cdots(n^3+1)$ is not a cube

TL;DR

This paper proves that the product of the sequence (k^3+1) for k from 1 to n is never a perfect cube for any positive integer n, resolving a specific number theory conjecture.

Contribution

It provides a rigorous proof that the sequence defined by the product of (k^3+1) is never a perfect cube, a novel result in number theory.

Findings

C_n is never a perfect cube for any positive integer n

The sequence (k^3+1) products do not produce cubes

The proof settles a conjecture in number theory

Abstract

For a positive integer $n,$ define $$C_n=\prod_{k=1}^n(k^3+1).$$ In this paper we prove that there are no cubes in the integer sequence $C_n,~n=1,2,\cdots.$