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

Simon Plouffe simon.plouffe at gmail.com
Fri Apr 6 08:10:36 CEST 2012

```
Yes, it is the number : ln(2+3^(1/2))/4, I have it in my
tables too. identify(%,all) of Maple finds it as well.

at least a 15 digits match.

best regards,
simon plouffe

Le 06/04/2012 06:38, Robert Munafo a écrit :
> 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
>>