- I'm pretty sure this has been looked at before, as it's so simple, but it's sometimes fun to resurrect old ideas. In particular when there's plenty of room for both carbon-based and silicon-based attacks on a puzzle.

Consider an ordered n-tuple V_0 of primes.

Define V_{i+1}, if it exists, to be V_i with the first element removed, and a final element added with value equal to the sum of all n primes in V_i. If that final element would be composite, then V_{i+1} does not exist.

e.g. with n=3, let V_0 = <3,3,5>

V_1 = <3,5,11>

V_2 = <5,11,19>

V_3 does not exist, as 5+11+19=35 is composite

Now here are the puzzles:

1) With n=3, is it possible for a sequence V_i to be infinite in length.

a) if so - what's the longest you can find?

b) if not, what's the maximum length, and why? What V_0 leads to the smallest final prime?

2) As above, but with n=5

3) As above, but with n=7

4) Generalise, what do you predict for other n?

Enjoy!

Phil

--

() ASCII ribbon campaign () Hopeless ribbon campaign

/\ against HTML mail /\ against gratuitous bloodshed

[stolen with permission from Daniel B. Cristofani] > Consider an ordered n-tuple V_0 of primes.

1) With n=3, is it possible for a sequence V_i to be infinite in length.

> Define V_{i+1}, to be V_i with the first element removed, and a final

> element added with value equal to the sum of all n primes in V_i. If that

> final element would be composite, then V_{i+1} does not exist.

> (...)

>

>

no,

> a) if so - what's the longest you can find?

examples with length=4 are

>

[61, 67, 71], [1709, 1721, 1723], [2371, 2377, 2381], [2543, 2549, 2551],

[4021, 4027, 4049], [5443, 5449, 5471], [5827, 5839, 5843] ,...

(here always [p(k),p(k+1),p(k+2)], I know this is not needed).

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

what is the "final prime" ?

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

> 2) As above, but with n=5

It seems that in these cases the maximum length is always 5.

> 3) As above, but with n=7

> 4) Generalise, what do you predict for other n?

>

But I may be wrong...

Maximilian

[Non-text portions of this message have been removed]- Maximilian Hasler wrote:
> b) if not, what's the maximum length, and why? What V_0 leads to the

You forgot there is one prime which is 0 mod 3.

>> 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

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 - --- 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

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

> > 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

0,1,1 => 1,1,2

0,0,1 => 0,1,1

> what is the "final prime" ?

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

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

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

So you don't think the 5-tuplet

> > 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...

[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