A note on Sierpiński's problem related to triangular numbers
Colloquium Mathematicum, Tome 117 (2009) no. 2, pp. 165-173
Cet article a éte moissonné depuis la source Institute of Mathematics Polish Academy of Sciences
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.
Keywords:
system equations quad quad where triangular number has infinitely many solutions integers moreover system has rational three parameter solution using result system quad quadt quad has infinitely many rational two parameter solutions
Affiliations des auteurs :
Maciej Ulas 1
@article{10_4064_cm117_2_2,
author = {Maciej Ulas},
title = {A note on {Sierpi\'nski's} problem related to triangular numbers},
journal = {Colloquium Mathematicum},
pages = {165--173},
year = {2009},
volume = {117},
number = {2},
doi = {10.4064/cm117-2-2},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.4064/cm117-2-2/}
}
Maciej Ulas. A note on Sierpiński's problem related to triangular numbers. Colloquium Mathematicum, Tome 117 (2009) no. 2, pp. 165-173. doi: 10.4064/cm117-2-2
Cité par Sources :