An unexpected meeting between the $P^{3}_{1}$-set and the cubic-triangular numbers

A set of $m$ positive integers $\{x_{1},\ldots,x_{m}\}$ is called a $P^{3}_{1}$-set of size $m$ if the product of any three elements in the set increased by one is a cube integer. A $P^{3}_{1}$-set $S$ is said to be extendible if there exists an integer $y\not\in S$ such that $S\cup\{y\}$ still a $P^{3}_{1}$-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 $P^{3}_{1}$-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.