Interesting Esoterica

2178 And All That

Article by Sloane, NJA
  • Published in 2013
  • Added on
In the collection
For integers g >= 3, k >= 2, call a number N a (g,k)-reverse multiple if the reversal of N in base g is equal to k times N. The numbers 1089 and 2178 are the two smallest (10,k)-reverse multiples, their reversals being 9801 = 9x1089 and 8712 = 4x2178. In 1992, A. L. Young introduced certain trees in order to study the problem of finding all (g,k)-reverse multiples. By using modified versions of her trees, which we call Young graphs, we determine the possible values of k for bases g = 2 through 100, and then show how to apply the transfer-matrix method to enumerate the (g,k)-reverse multiples with a given number of base-g digits. These Young graphs are interesting finite directed graphs, whose structure is not at all well understood.

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journal
arXiv preprint arXiv:1307.0453

BibTeX entry

@article{Sloane2013,
	author = {Sloane, NJA},
	journal = {arXiv preprint arXiv:1307.0453},
	title = {2178 And All That},
	url = {http://arxiv.org/abs/1307.0453 http://arxiv.org/pdf/1307.0453v4},
	year = 2013,
	archivePrefix = {arXiv},
	eprint = {1307.0453},
	primaryClass = {math.NT},
	abstract = {For integers g >= 3, k >= 2, call a number N a (g,k)-reverse multiple if the
reversal of N in base g is equal to k times N. The numbers 1089 and 2178 are
the two smallest (10,k)-reverse multiples, their reversals being 9801 = 9x1089
and 8712 = 4x2178. In 1992, A. L. Young introduced certain trees in order to
study the problem of finding all (g,k)-reverse multiples. By using modified
versions of her trees, which we call Young graphs, we determine the possible
values of k for bases g = 2 through 100, and then show how to apply the
transfer-matrix method to enumerate the (g,k)-reverse multiples with a given
number of base-g digits. These Young graphs are interesting finite directed
graphs, whose structure is not at all well understood.},
	urldate = {2013-11-20},
	collections = {Easily explained}
}