Descending Dungeons and Iterated Base-Changing

• Published in 2006
• Added on 2022-02-28

For real numbers a, b> 1, let as a_b denote the result of interpreting a in base b instead of base 10. We define ``dungeons'' (as opposed to ``towers'') to be numbers of the form a_b_c_d_..._e, parenthesized either from the bottom upwards (preferred) or from the top downwards. Among other things, we show that the sequences of dungeons with n-th terms 10_11_12_..._(n-1)_n or n_(n-1)_..._12_11_10 grow roughly like 10^{10^{n log log n}}, where the logarithms are to the base 10. We also investigate the behavior as n increases of the sequence a_a_a_..._a, with n a's, parenthesized from the bottom upwards. This converges either to a single number (e.g. to the golden ratio if a = 1.1), to a two-term limit cycle (e.g. if a = 1.05) or else diverges (e.g. if a = frac{100{99).

Links

• http://arxiv.org/abs/math/0611293v3
• http://arxiv.org/pdf/math/0611293v3

Other information

BibTeX entry

@article{DescendingDungeonsandIteratedBaseChanging,
	key = {DescendingDungeonsandIteratedBaseChanging},
	type = {article},
	title = {Descending Dungeons and Iterated Base-Changing},
	author = {David Applegate and Marc LeBrun and N. J. A. Sloane},
	abstract = {For real numbers a, b> 1, let as a{\_}b denote the result of interpreting a in
base b instead of base 10. We define ``dungeons'' (as opposed to ``towers'') to
be numbers of the form a{\_}b{\_}c{\_}d{\_}...{\_}e, parenthesized either from the bottom
upwards (preferred) or from the top downwards. Among other things, we show that
the sequences of dungeons with n-th terms 10{\_}11{\_}12{\_}...{\_}(n-1){\_}n or
n{\_}(n-1){\_}...{\_}12{\_}11{\_}10 grow roughly like 10^{\{}10^{\{}n log log n{\}}{\}}, where the
logarithms are to the base 10. We also investigate the behavior as n increases
of the sequence a{\_}a{\_}a{\_}...{\_}a, with n a's, parenthesized from the bottom upwards.
This converges either to a single number (e.g. to the golden ratio if a = 1.1),
to a two-term limit cycle (e.g. if a = 1.05) or else diverges (e.g. if a =
frac{\{}100{\{}99).},
	comment = {},
	date_added = {2022-02-28},
	date_published = {2006-04-10},
	urls = {http://arxiv.org/abs/math/0611293v3,http://arxiv.org/pdf/math/0611293v3},
	collections = {attention-grabbing-titles,easily-explained,fun-maths-facts,integerology},
	url = {http://arxiv.org/abs/math/0611293v3 http://arxiv.org/pdf/math/0611293v3},
	year = 2006,
	urldate = {2022-02-28},
	archivePrefix = {arXiv},
	eprint = {math/0611293},
	primaryClass = {math.NT}
}