A note on Sierpiński's problem related to triangular numbers

Maciej Ulas. "A note on Sierpiński's problem related to triangular numbers." Colloquium Mathematicae 117.2 (2009): 165-173. <http://eudml.org/doc/283925>.

@article{MaciejUlas2009,
abstract = {We show that the system of equations $t_\{x\} + t_\{y\} = t_\{p\}, t_\{y\} + t_\{z\} = t_\{q\}, t_\{x\} + t_\{z\} = t_\{r\}$, 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 $t_\{x\} + t_\{y\} = t_\{p\}, t_\{y\} + t_\{z\} = t_\{q\}, t_\{x\} + t_\{z\} = t_\{r\}, t_\{x\} + t_\{y\}+t_\{z\} = t_\{s\}$ has infinitely many rational two-parameter solutions.},
author = {Maciej Ulas},
journal = {Colloquium Mathematicae},
keywords = {sums of triangular numbers; rational points},
language = {eng},
number = {2},
pages = {165-173},
title = {A note on Sierpiński's problem related to triangular numbers},
url = {http://eudml.org/doc/283925},
volume = {117},
year = {2009},
}

TY - JOUR
AU - Maciej Ulas
TI - A note on Sierpiński's problem related to triangular numbers
JO - Colloquium Mathematicae
PY - 2009
VL - 117
IS - 2
SP - 165
EP - 173
AB - We show that the system of equations $t_{x} + t_{y} = t_{p}, t_{y} + t_{z} = t_{q}, t_{x} + t_{z} = t_{r}$, 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 $t_{x} + t_{y} = t_{p}, t_{y} + t_{z} = t_{q}, t_{x} + t_{z} = t_{r}, t_{x} + t_{y}+t_{z} = t_{s}$ has infinitely many rational two-parameter solutions.
LA - eng
KW - sums of triangular numbers; rational points
UR - http://eudml.org/doc/283925
ER -