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".
Can you confirm if the Jobling's
is the 5th member of the d->(d)d
sequence, or to get the least if this is not the case?
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
12815137 -> 14641251949 -> 116361614251811681 ->
11369361361164251641136641 -> 11936819361936113616425136161193636161
Then, this is the corresponding sequence:
2, 13, 13, 131, 2111, 12815137, ?
Question 3: Can you extend the d->d2 sequence?
III.- The d->dd 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->dd
This is the corresponding sequence:
2, 13, 367, 8071171, ?, ...
367-> 2746656823543 -> 4823543256466564665631254665616777216427312525627
-> 167772161823543118235431 ->
Question 4: Can you extend the d->dd sequence?
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."
Eleven years later (Set, 2011) Giovanni Resta found that the Jobling's result for Q1 was wrong and more issues: