# A note on Sierpiński's problem related to triangular numbers

Colloquium Mathematicae (2009)

- Volume: 117, Issue: 2, page 165-173
- ISSN: 0010-1354

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topMaciej 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 -

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