prime twin pairs yielding others

[Neil Fernandez]

Many thanks Rick and Don.

After posting, I realised that the first question is fairly easy. All
terms in the sequence, other than the first, *are* congruent to 29 (mod
30).

Proof:

for integers greater than 30, only those with the following congruence
(mod 30) are capable of forming twin pairs.

smaller greater
11      13
17      19
29      1

(5,7) is ruled out because the smaller number is divisible by 5, and
(23,25) is ruled out because the greater number is similarly divisible.

If primes with congruences (11,13) formed a twin pair which yielded
another, then the primes in the latter pair would have congruences
(23,25). This is impossible, so (11,13) can be struck off the list.
Similarly, if primes with congruences (17,19) formed a twin pair which
yielded another, then the primes in the latter pair would have
congruences (5,7). This is also impossible, so (17,19) can be struck off
the list too.

The only case remaining on the list is (29,1). So the smaller prime
(>30) in any twin pair that yields another twin pair must be congruent
to 29 (mod 30). We know that the second prime on the list is 29, so this
is true for all primes on the list after the first.



The second question is harder :-) I would conjecture that for positive
integer n, it may be impossible to prove that the primorials do not
include infinitely many numbers which are separators of twin pairs
yielding 'twin pair chains' of length n.

Neil
-- 
Neil Fernandez