Interesting Esoterica

When are Multiples of Polygonal Numbers again Polygonal Numbers?

Article by Jasbir S. Chahal and Nathan Priddis
  • Published in 2018
  • Added on
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Euler showed that there are infinitely many triangular numbers that are three times another triangular number. In general, as we prove, it is an easy consequence of the Pell equation that for a given square-free m > 1, the relation D = mD' is satisfied by infinitely many pairs of triangular numbers D, D'. However, due to the erratic behavior of the fundamental solution to the Pell equation, this problem is more difficult for more general polygonal numbers. We will show that if one solution exists, then infinitely many exist. We give an example, however, showing that there are cases where no solution exists. Finally, we also show in this paper that, given m > n > 1 with obvious exceptions, the simultaneous relations P = mP', P = nP" has only finitely many possibilities not just for triangular numbers, but for triplets P, P', P" of polygonal numbers.

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BibTeX entry

@article{WhenareMultiplesofPolygonalNumbersagainPolygonalNumbers,
	title = {When are Multiples of Polygonal Numbers again Polygonal Numbers?},
	author = {Jasbir S. Chahal and Nathan Priddis},
	url = {http://arxiv.org/abs/1806.07981v1 http://arxiv.org/pdf/1806.07981v1},
	urldate = {2018-06-25},
	year = 2018,
	abstract = {Euler showed that there are infinitely many triangular numbers that are three
times another triangular number. In general, as we prove, it is an easy
consequence of the Pell equation that for a given square-free m > 1, the
relation D = mD' is satisfied by infinitely many pairs of triangular numbers D,
D'. However, due to the erratic behavior of the fundamental solution to the
Pell equation, this problem is more difficult for more general polygonal
numbers. We will show that if one solution exists, then infinitely many exist.
We give an example, however, showing that there are cases where no solution
exists. Finally, we also show in this paper that, given m > n > 1 with obvious
exceptions, the simultaneous relations P = mP', P = nP" has only finitely many
possibilities not just for triangular numbers, but for triplets P, P', P" of
polygonal numbers.},
	comment = {},
	archivePrefix = {arXiv},
	eprint = {1806.07981},
	primaryClass = {math.NT},
	collections = {easily-explained}
}