(2^n+1)/3
Richard Guy
rkg at cpsc.ucalgary.ca
Fri May 20 18:02:16 CEST 2005
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
>
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