[seqfan] Prime concatenations to erase

[Eric Angelini]

Hello SeqFans,
Erase all pairs of integers in S that,
concatenated, produce a prime;
the remaining integers are S itself.

S= 1,9,1,2,3,9,1,2,4,21,3,9,1,2,4,5,23,21,
3,9,1,2,4,5,6,7,23,21,3,9,1,2,4,5,6,8,
11,7,23,21,3,9,1,2,4,5,6,8,10,13,11,7,
23,21,3,9,1,2,4,5,6,8,10,12,17,13,11,
7,23,21,3,9,1,2,4,5,6,8,10,12,14,27,
17,13,11,7,23,21,3,9,1,2,4,5,6,8,10,
12,14,15,31,27,17,13,11,7,23,21,3,9,
1,2,4,5,6,8,10,12,14,15,16,37,31,27,
17,13,11,7,23,21,3,9,1,2,4,5,6,8,10,
12,14,15,16,18,47,37,31,27,17,13,11,
7,23,21,3,9,1,2,4,5,6,8,10,12,14,15,
16,18,20,29,47,37,31,27,17,13,11,7,
23,21,3,9,1,2,4,5,6,8,10,12,14,15,16,
18,20,22,43,29,47,37,31,27,17,13,11,
7,23,21,3,9,1,2,4,5,6,8,10,12,14,15,
16,18,20,22,24,...

Example:

Erase the pair 1 and 9 as 19 is prime
then erase 2 and 3 as 23 is prime
then erase 4 and 21 as 421 is prime
then erase 5 and 23 as 523 is prime
... etc.

The building method is (as usual)
pyramidal:
                                 1
                              9,1,2
                           3,9,1,2,4
                      21,3,9,1,2,4,5
                 23,21,3,9,1,2,4,5,6
              7,23,21,3,9,1,2,4,5,6,8
         11,7,23,21,3,9,1,2,4,5,6,8,10
    13,11,7,23,21,3,9,1,2,4,5,6,8,10,12
17,13,11,7,23,21,3,9,1,2,4,5,6,8,10,12,14,...

etc.

We see that we only have to check that:
[A] the new integer starting a raw
doesn't produce a prime with the
term coming next (9 together with 1 is
ok as 91 is not prime);
[B] the integer ending a raw:
1) doesn't produce a prime with the integer coming just before it (1 and 2
are ok as 12 is not prime);
2) does produce a prime with the integer
starting the next raw (2 and 3 are ok
as 23 is prime).

S was always extended with the smallest
integer not leading to a contradiction.
(this explains why, after the initial "1",
no other integer is available before 9:
1 and the integer we want must form
a prime, so 2,4,5,6 and 8 are forbidden;
but 3 and 7 would form 31 and 37 which are primes too (forbidden); we
then take 9)

Best,
É.