2178 And All That
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.
- Publication:
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arXiv e-prints
- Pub Date:
- July 2013
- DOI:
- 10.48550/arXiv.1307.0453
- arXiv:
- arXiv:1307.0453
- Bibcode:
- 2013arXiv1307.0453S
- Keywords:
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- Mathematics - Number Theory;
- Mathematics - Combinatorics;
- 11A63
- E-Print:
- 22 pages, 16 figures, one table. July 4 2013: corrected typo in table, added conjectures about particular graphs. Sept. 24 2013: corrected typos, added conjectures and theorems. Oct 13 2013: minor edits