A note on Sierpi\'{n}ski problem related to triangular numbers

TL;DR

This paper investigates systems of equations involving triangular numbers, demonstrating the existence of infinitely many integer and rational solutions, including parametric solutions, thus advancing understanding of related Diophantine problems.

Contribution

It establishes the existence of infinitely many solutions and provides explicit parametric solutions for systems involving sums of triangular numbers.

Findings

Infinitely many integer solutions to the system of triangular number equations.

Existence of rational three-parametric solutions for the basic system.

Infinite rational two-parametric solutions when an additional sum condition is included.

Abstract

In this note we show that the system of equations t_{x}+t_{y}=t_{p},\quad t_{y}+t_{z}=t_{q},\quad t_{x}+t_{z}=t_{r}, where $t_{x}=x(x+1)/2$ is a triangular number, has infinitely many solutions in integers. Moreover we show that this system has rational three-parametric solution. Using this result we show that the system t_{x}+t_{y}=t_{p},\quad t_{y}+t_{z}=t_{q},\quad t_{x}+t_{z}=t_{r},\quad t_{x}+t_{y}+t_{z}=t_{s} has infinitely many rational two-parametric solutions.