[seqfan] Re: Prime concatenations to erase

[hv at crypt.org]

Hi Eric, can you tighten up the definition to explain why the following
two sequences do not qualify? They both lie lexically before your example.

1, 2, 1, 2, 1, 2, ... (nothing erased)
1, 1, 1, 1, 1, 1, ... (everything erased, at least initially)

It also isn't clear to me what should happen with overlapping primes,
ie whether 2, 1, 3, 7, 2 is erased down to 2, 7, 2 or down to 2, 2.

Hugo

Eric Angelini <Eric.Angelini at kntv.be> wrote:
: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.
:
:
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