Problems & Puzzles: Puzzles

Puzzle 112. Automorphic primes

I.- The d->(d)_d transformation

A few days ago G. L. Honaker, Jr. was re-exploring the Conway's Look and say - sequence and had the idea of create sequences of primes by the simple procedure of replacing digits using certain specific transformations.

At the beginning he selected the following procedure: replace the digit 'd' with d digits 'd'. Do this for all the digits of the current number.

Then he started asking & getting primes (non trivial) that generates other primes?

If we restrict to only one iteration of the procedure the answer is 31:

31 --> 3331

He also found the least prime that provides a sequence of 3 primes, applying twice the procedure:

641--> 66666644441--> (6)₃₆(4)₁₆1

I (C.R.) found the least prime (12422153) that starts a sequence of 4 primes and Paul Jobling has produced a prime (142112242123, not necessarily the least) that starts a sequence of 5 primes

Then we have the following sequence: 

2, 31, 641, 12422153, 142112242123 , ... 

Whose description is:

"The least prime that starts a sequence of k primes using recursively the d->(d)_d transformation k-1 times". 

Questions:

1. Can you confirm if the Jobling's prime (142112242123) is the 5th member of the d->(d)_d sequence, or to get the least if this is not the case?
2. Can you get the 6th member of the sequence d->(d)_d ?

II.- The d->d² transformation

More recently G. L. Honaker asked for the corresponding sequences for the following transformation d->d². Himself sent the 131 -> 191 -> 1811 -> 16411 (4 primes) example. I produced the following two examples:

a) 2111-> 4111-> 16111-> 136111-> 1936111 (5 primes) and

b) 12815137 -> 14641251949 -> 116361614251811681 -> 11369361361164251641136641 -> 11936819361936113616425136161193636161 ->
1181936641819361819361193613616425193613611819369361361 (6 primes)

Then, this is the corresponding sequence:

2, 13, 13, 131, 2111, 12815137, ?

Question 3: Can you extend the d->d² sequence?

III.- The d->d^d transformation

Very naturally I extended the idea of G. L. Honaker making the power of the digit d to be the same digit d, that is to say, d->d^d  

This is the corresponding sequence:

2, 13, 367, 8071171, ?, ...

In this transformation:
13 -> 127

367-> 2746656823543 -> 4823543256466564665631254665616777216427312525627

and

8071171 -> 167772161823543118235431 ->
146656823543823543823543414665611677721642731252562711167772164273125256271 ->
125646656466563125466561677721642731252562716777216427312525627167772164273/
12525627256125646/656466563125466561146656823543823543823543414665625648235/
43271431254312546656482354311146656/823543823543823543414665625648235432714/
3125431254665648235431

Question 4: Can you extend the d->d^d sequence?

Solution

As per Question 2 Felice Russo wrote (20/11/2000): "After 1 week of search I was not able to find any further solution between 12815137 and 63*10^8."

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