Problems & Puzzles: Puzzles

Puzzle 126. Some conspicuous probable primes (*)

1.- Find the first probable prime in the following sequences:

1.a) 1, 122, 122333, 1223334444, etc..

1.b) 1, 221, 333221, 4444333221, etc.

2.- Find the fourth probable prime in the following generalized palindromic (**) sequence:

11,1221,122333221, 1223334444333221, etc.

(primes, as always, are in blue)

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Notes:
(*) I'm posting this 3 questions with the hope of producing a probable prime that may serve to get another record for the Titanix code. This requires that at least one of the probable primes asked has less than 4000 digits and more than 3106 digits. Why? The current record for this powerful tool - Titanix - is the now prime number (348^1223-1)/347, 3106 decimal digits, proved to be prime very recently by Giovanni & Marco La Barbera. If all of these 3 probable primes are too larger than 4000 digits, then the task will have to be solved in the near? future...). If all the asked probable primes are out of the suggested range maybe you would be so kind to submit another digital-conspicuous probable prime & candidate to be the record for the Titanix code, with a quantity of digits in the proper range.

(**) A generalized palindrome sequence is a concatenation of numbers abc...xyz such that a=z, b=y, c=x, etc. The nut or center of the number may be empty (for a gp with an even quantity of concatenated numbers) or an arbitrary number (for a gp with an odd quantity of concatenated numbers).  If and only if every concatenated number in the gp is palindrome then, gp is also a classical or real palindrome number.

J. K. Andersen wrote:

I tested the 3 sequences with PrimeForm/GW. They grow so quickly that it
would have required a lot of luck to get a prp in Titanix/Primo record range. No primes or prp's were found.

1.a) No primes in the first 708 terms. Term 709 has 750090 digits.

1.b) No primes in the first 599 terms. Term 600 has 535905 digits.

2. No primes in term 6 to term 471. Term 472 has 658362 digits.
I think the first term should be 1 and not 11.
Then there are only two small primes: Term 3 and term 5.

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