(2^n+1)/3

[Richard Guy]

Amer Math Monthly, 96(1989) 125:

`The New Mersenne Conjecture'
(Bateman-Selfridge-Wagstaff) states
that:

If two of the following statements about
an odd positive integer  p  are true, then
the third one is also true

(a)  p = 2^k \pm 1  or  p = 4^k \pm 3

(b)  M_p  is prime

(c)  (2^p + 1)/3  is prime.

[It's not necessary to assume that  p
is prime.  M_p is the Mersenne number
2^p - 1]             R.

On Fri, 20 May 2005, Gottfried Helms wrote:

> 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?
>>
> An addition to my previous post:
>
> You can rewrite the 3rd part of the "bateman-..."-conjecture, which
> states
>   3)  (2^p+1)/3 = q    is prime
>
> into
>    2^p + 1  = 3*q
>
> and the cyclic length of 2^p+1 seems always to be
>
>    L2(2^p+1) = 2*p
>
> so that
>
>    2*p = l2(3*q)
>
> which, for the multiplicity in my assumption, means
>
>    2*p = lcm(l2(3),l2(q)) = lcm(2,l2(q))
>
> This is immediately true for a mersenne-prime q, where
>  q = 2^p-1
>
> since then
>  L2(q) = L2(2^p-1) = p
>
> and then
>
>   2*p = lcm(2,p) = 2*p
>
> Possibly using the 1st condition of the "Bateman..."-conjecture
> one can narrow this down to a *required* condition by just algebraic
> manipulation of the above formulas - always on the basis of
> the -assumed- uniqueness cycle-length of mersenne-primes.
>
> I'll try with this a bit today.
>
>
> Gottfried Helms
>