(2^n+1)/3

[Gottfried Helms]

Am 20.05.05 08:50 schrieb kohmoto:
> 
>     Hello, Seqfans
>     Once I realized that if M_n is a Mersenne prime then (M_n+2)/3 is
> also prime. 2<n
>     And I knew that it is called "Bateman and Shefridge and Wagstaff's
> conjecture ".
>     Does anyone know the exact description of it?
>  
>     %I A000001
>     %S A000001 3, 11, 43, 2731, 43691
>     %N A000001 (M_n+2)/3. Where M_n is Mersenne prime 2^n-1.
>     %C A000001 If "Bateman and Shefridge and Wagstaff's conjecture " is
> true, then all terms of the sequence becomes primes.
>      %O A000001     3,1 
>     %Y A000001    A000668
>     %K A000001    nonn
>     %A A000001    Yasutoshi Kohmoto (zbi74583 at boat.zero.ad.jp)
>  
> 
>  
Hi Yasutoshi -

 I apologize for my previous postings, they were not much helpful
 and may have seem to be not focused to your question, when I reread
 them again.
 What I was trying was to be helpful with a line of arguments,
 which were related to

  http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A007733

 the sequence of the residues of n'th powers of 2 (modulo n) , which seemed
 promising.
 On the net a few days ago I found a fully explicite discussion of
 that idea, applied to some number-theory-problems in the same manner, as I
 tried to develop recently.

 see http://www.dybot.com/numbers/sqfree.htm

 Unfortunately, this path does actually *not* give a decisive help to get
 nearer to a confirmation or disconfirmation of your observation,- conversely
 your (resp. the Batesman-et-al-) conjecture is independent to the current
 observations which were made when applying and analyzing that (unfortunately
 incomplete) apparatus:
 it is equivalent to a question whether unique *even* cycle-lengthes exist,
 (which are also connected to mersenne-primes).

 Nearly all that relevant observations and current assumptions still need
 rigid proofs, so that optimistic postings of mine were also misleading.

 Kind regards -

Gottfried Helms