Problems & Puzzles: Puzzles

Puzzle 999. In Memoriam to John Horton Conway*

Starting the year 1999 I posted the Puzzle 36, related to "Sequences of Descriptive terms".

In that very year, and no later than 23/09/01, Carlos Rivera, Mike Keith and Walter Schneider found in total seven examples of sequences of descriptive all prime six terms:

 Starting term Found by 233 CR 120777781 CR 402266411 CR 1171465511 MK 1623379207 MK 1955771683 MK 5558223787 WS

The smallest example, all prime six terms self descriptive sequence:

233, 1223, 112213, 21221113, 1211223113, 11122122132113. (CBRF, 22/1/99)

Mike Keith contributed also with the estimations about how many self-descriptive sequences one should have to examine (of length N) in order to find one that is all primes. The estimation made use of one Conway's theorem, which says that self-descriptive sequences grow (in terms of the number of digits) proportional to 1.3^n.

On 29/1/99 the Keith's estimations predicted that "the first length-7 chain will occur around 10^11, and the first length-8 chain (which I can barely imagine ever finding!) around 10^13.  But, of course, we may be lucky and find one earlier."

On 25/09/01 Walter Schneider found the first seven terms sequence in total accordance with the Keith's prediction:

"...One sequence of length 7 starting at 19.972.667.609 (found today at 25.09.2001)..."

Perhaps is time now (almost twenty years later) to find the one self-descriptive sequence of eight all prime terms.

Q1. Send your smallest solution (self-descriptive sequence of eight all prime terms.).

If we add the condition that the first term of the sequence must be a palprime, the record is from Tiziano Mosconi: a sequence of five all prime terms:

1344409044431  , 111334101910341311  ,   3123141110111911101314111321  ,
1311121311143110311931101113111431131211  ,
111331121113311413211013211913211031133114132113111221

Q2. Send your smallest solution (self-descriptive sequence of six all prime terms, the first one, a palprime).

_________
*Who passed away  on the 11th April in New Brunswick, New Jersey, at the age of 82

During the week 2-9, May 2020, contributions came from Giovanni Resta,

***

Giovanni wrote:

Q1. The first sequence of 8 primes starts at 75022592087629.

75022592087629,

17151022151912101817161219,

11171115111022111511191112111011181117111611121119,

31173115311022311531193112311031183117311631123119,

13211713211513211022132115132119132112132110132118132117132116132112132119,

11131221171113122115111312211022111312211511131221191113122112111312211011/
131221181113122117111312211611131221121113122119,

31131122211731131122211531131122211022311311222115311311222119311311222112/
311311222110311311222118311311222117311311222116311311222112311311222119,

13211321322117132113213221151321132132211022132113213221151321132132211913/
21132132211213211321322110132113213221181321132132211713211321322116132113/
2132211213211321322119

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