# Descending Dungeons and Iterated Base-Changing

• Published in 2006
In the collections
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).

## Other information

key
DescendingDungeonsandIteratedBaseChanging
type
article
2022-02-28
date_published
2006-07-11

### 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 = {},
}