A note on Sierpiński's problem related to triangular numbers

Volume 117 / 2009

Maciej Ulas Colloquium Mathematicum 117 (2009), 165-173 MSC: 11D41, 11D72, 11D25. DOI: 10.4064/cm117-2-2


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.


  • Maciej UlasInstitute of Mathematics
    Jagiellonian University
    Łojasiewicza 67
    30-348 Kraków, Poland

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