A note on Sierpiński's problem related to triangular numbers
Tom 117 / 2009
Colloquium Mathematicum 117 (2009), 165-173
MSC: 11D41, 11D72, 11D25.
DOI: 10.4064/cm117-2-2
Streszczenie
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.
