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

Maximilian Hasler maximilian.hasler at gmail.com
Fri Apr 6 16:41:54 CEST 2012

```For what it's worth, the explicit expression for the n-th term of the
sequence (and the related constant) is well known since "ever", and
various forms are given in A003010.

The interest of the test is that the computation **mod Mp** allows to
decide about primality, the explicit terms themselves (without mod Mp)
are of no practical interest in view of their size (note that not just
the first 10, 100 or 1000 terms are needed, but at present one is
interested in  n ~ Mp ~ 2^p with p ~ 5x10^7
and a(n) = (2+sqrt(3))^(2^n)  would have 0.597 * 2^2^5e7 digits...

But this does not mean that such constants should not be added to OEIS
which is still much less complete than Simon Plouffe's tables, btw
2+sqrt(3) = A019973 is already there since 1996 at least.

Maximilian

On Fri, Apr 6, 2012 at 2:10 AM, Simon Plouffe <simon.plouffe at gmail.com> wrote:
>
>
> 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
>>>
>>
>>
>>
>> LucLeh ~
>>>
>>> 0.329239474231204177156261586826992111006745492821106086516800... and
>>> actually LucLeh = acosh(sqrt(2+sqrt(2+4))/2)
>>>
>>> [...]
>>>
>
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