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

Colloquium Mathematicae (2009)

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

top

## Abstract

top
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 ${t}_{x}=x\left(x+1\right)/2$ 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.

## How to cite

top

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 -

## NotesEmbed?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.