Permutations: Adjacent Elements Sum to Primes

[T. D. Noe]

At 4:18 PM -0500 3/26/05, Edwin Clark wrote:
>On Sat, 26 Mar 2005, James Buddenhagen wrote:
>>
>> Yes. A051239 is your sequence, each term divided by 2, and starting at 1.
>>
>
>This and sequence
>
>ID Number: A036440
>URL:       http://www.research.att.com/projects/OEIS?Anum=A036440
>
>suggests that there may be a permutation p of the natural numbers such
>that p(i) + p(i+1) is prime for all i. Is anything known about this?

There are a few references in Guy's UPINT section C1.  No solutions!

It appears that we can find a permutation in which the even (or odd)
numbers are  in order.  Can we always choose a permutation that yields only
twin primes?

Tony