A note on Sierpiński's problem related to triangular numbers

Maciej Ulas

Maciej Ulas

Colloquium Mathematicae, Tome 116 (2009), p. 165-173 / Harvested from The Polish Digital Mathematics Library

Résumé

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.

Publié le : 2009-01-01

Zbl 1214.11039

EUDML-ID : urn:eudml:doc:283925

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     author = {Maciej Ulas},
     title = {A note on Sierpi\'nski's problem related to triangular numbers},
     journal = {Colloquium Mathematicae},
     volume = {116},
     year = {2009},
     pages = {165-173},
     zbl = {1214.11039},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm117-2-2}
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Maciej Ulas. A note on Sierpiński's problem related to triangular numbers. Colloquium Mathematicae, Tome 116 (2009) pp. 165-173. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm117-2-2/