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## Re: [PrimeNumbers] summing primes to yield new primes

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• ... You forgot there is one prime which is 0 mod 3. Phil s example was V_0 = so two 3 s is clearly also allowed. [3, 3, 11] gives length 6 with the
Message 1 of 4 , Jun 19, 2012
Maximilian Hasler wrote:
> b) if not, what's the maximum length, and why? What V_0 leads to the
>> smallest final prime?
>>
> the maximum length is 4, as you can see when you look at the sequence V_i
> mod 3:
> 1,1,1 => 1,1,0
> 2,2,2 => 2,2,0
> 1,1,2 => 1,2,1
> 1,2,1 => 2,1,1
> 2,1,1 => 1,1,1
> 1,2,2 => 2,2,2
> 2,1,2 => 1,2,2
> 2,2,1 => 2,1,2

You forgot there is one prime which is 0 mod 3.
Phil's example was V_0 = <3,3,5> so two 3's is clearly also allowed.
[3, 3, 11] gives length 6 with the added terms 17, 31, 59, 107, 197.

--
Jens Kruse Andersen
• ... You re permitted to start with 3s, so that can be extended by 2. 0,1,1 = 1,1,2 0,0,1 = 0,1,1 ... (2,2,6) doesn t exist, so (2,2,2) doesn t lead there. In
Message 2 of 4 , Jun 19, 2012
--- On Tue, 6/19/12, Maximilian Hasler <maximilian.hasler@...> wrote:
> b) if not, what's the maximum length, and why? What V_0 leads to the
> > smallest final prime?

> the maximum length is 4, as you can see when you look at the
> sequence V_i
> mod 3:
> 1,1,1 => 1,1,0
> 2,2,2 => 2,2,0
> 1,1,2 => 1,2,1
> 1,2,1 => 2,1,1
> 2,1,1 => 1,1,1
> 1,2,2 => 2,2,2
> 2,1,2 => 1,2,2
> 2,2,1 => 2,1,2

You're permitted to start with 3s, so that can be extended by 2.
0,1,1 => 1,1,2
0,0,1 => 0,1,1

> what is the "final prime" ?
> (2,2,2) leads to (2,2,6), so the final prime is 2 ?

(2,2,6) doesn't exist, so (2,2,2) doesn't lead there.

In the <3,3,5> example I gave, 19 is the final prime, as no other primes were added after it.

> >  2) As above, but with n=5
> > 3) As above, but with n=7
> > 4) Generalise, what do you predict for other n?
> >
> It seems that in these cases the maximum length is always
> 5. But I may be wrong...

So you don't think the 5-tuplet
[5, 83, 3, 7, 3]
would be extended in turn 8 times by the primes
[101, 197, 311, 619, 1231, 2459, 4817, 9437]
(and no, that's not the answer, merely illustrative)?

It seems you've made some assumptions that have stopped you from investigating every possibility.

Phil
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