A note on Sierpiński's problem related to triangular numbers

Maciej Ulas¹

¹ Institute of Mathematics Jagiellonian University Łojasiewicza 67 30-348 Kraków, Poland

Colloquium Mathematicum, Tome 117 (2009) no. 2, pp. 165-173.

Résumé

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 a rational three-parameter 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-parameter solutions.

Détail

DOI : 10.4064/cm117-2-2

Keywords: system equations quad quad where triangular number has infinitely many solutions integers moreover system has rational three parameter solution using result system quad quadt quad has infinitely many rational two parameter solutions