# [seqfan] Re: A (new) constant related to the Lucas-Lehmer-test. Is this worth an entry in OEIS?

Robert Munafo mrob27 at gmail.com
Fri Apr 6 06:38:48 CEST 2012

```Hi Gottfried,

I like those sorts of things and will probably add it to my numbers page
[1], but I was curious, so I did a Google search and found a possible
match, by an author that clearly has no idea of a connection (if any).

Your LucLeh constant with 15 digits (0.329239474231204) appears in a table
of coefficients in this paper: arxiv.org/pdf/cond-mat/0610690 (Table 1,
with n=4, rho=1). It is a 2008 paper by Yong Kong having to do with
arranging dimers on 2D rectangular lattices.

Apart from your own paper, and your own postings to mathforum.org and
sci.math, the above is the only match that Google gives to either of your
constants having more than 9 digits (I tried everything from 0.32923947
to 0.32923947423120417715, and the short ones are just coincidences. The
longest coincidental match for ELucLeh is also a 9-digit approximation,
1.38991066.)

Since the Kong match has 6 more digits than the best coincidence-match on
Google, I suspect it would be worth looking at that paper and try to see
what their formula is doing and how it might relate to yours.

- - -

Lesser things: suggestions for your paper...

Because I have RIES [2] and I'm lazy, I used it to get a "simpler" formula
for LucLeh, namely:

LucLeh = ln(2+sqrt(3)) / 4

which you might want to mention in your paper (for readers who dislike
acosh). I suppose it comes from an identity for arccos like that at [3].

I had a little trouble verifying the 60-digit
value 0.329239474231204177156261586826992111006745492821106086516800 for
the expression acosh(sqrt(2+sqrt(2+4))/2). My bc init file doesn't have an
acosh function so I subtracted cosh of your 60-digit value
from sqrt(2+sqrt(2+4))/2, and I got a difference of about 1.5344 x 10^-47.

- Robert

[1] http://mrob.com/pub/math/numbers.html

[2] http://mrob.com/pub/ries/index.html

[3]
http://en.wikipedia.org/wiki/Hyperbolic_function#Inverse_functions_as_logarithms

On Thu, Apr 5, 2012 at 17:09, Gottfried Helms <helms at uni-kassel.de> wrote:

> Just for my own experience with iterations of functions I fiddled with the
> Lucas-Lehmer-test for Mersenne-numbers, which is just an application of
> functional iteration, beginning at a fxied starting value.
>
> This lead to the finding of a new constant, which for what it is worth, I
> temporarily called the "Lucas-Lehmer-Constant" which "encodes" the
> Lucas-Lehmer-test in one number. [...]
>
>    http://go.helms-net.de/math/expdioph/lucasLehmer.pdf
>

LucLeh ~
> 0.329239474231204177156261586826992111006745492821106086516800... and
> actually LucLeh = acosh(sqrt(2+sqrt(2+4))/2)
>
> [...]
>

--
Robert Munafo  --  mrob.com