An unexpected meeting between the $P_1^3$-set and the cubic-triangular numbers

Abstract

A set of $m$ positive integers $\{x_1,\dots ,x_m\}$ is called a $P_1^3$-set of size $m$ if the product of any three elements in the set increased by one is a cube integer. A $P_1^3$-set $S$ is said to be extendible if there exists an integer $y\notin S$ such that $S\cup \{y\}$ still a $P_1^3$-set. Now, let consider the Diophantine equation $u(u+1)/2=v^3$ whose integer solutions produce what we called cubic-triangular numbers. The purpose of this paper is to prove simultaneously that the $P_1^3$-set $\{1,2,13\}$ is non-extendible and $n=1$ is the unique cubic-triangular number by showing that the two problems meet on the Diophantine equation $2x^3-y^3=1$ that we solve using $p$-adic analysis.