Problems & Puzzles: Puzzles

Puzzle 134.  The 1379-Carrousel-Primes

Let's ask for primes P that remains prime when the following set of transformations applies at the same time:

a) all the "1" digits transforms in "3" digits
b) all the "3" digits transforms in "7" digits
c) all the "7" digits transforms in "9" digits
d) all the "9" digits transforms in "1" digits
e) all the other digits (0, 2, 4, 5, 6 & 8), remain the same ('inactive' digits) 

Let's ask only for these primes P that remain prime under 3 successive applications of the complete before mentioned set of transformations.

Here are some easy examples:

I. The least carrousel prime with these properties (and at least on 'active' digit) is: 19 -> 31 -> 73 ->97. Incidentally this carrousel prime has all its digits 'active' (to be honest the least carrousel primes are 2 & 5)

II. The least carrousel palprime is: 131 -> 373 -> 797 -> 919. Incidentally this palprime has also all its digits 'active'.

III. The least carrousel prime that generates two couples of reversible primes is:157 -> 359 -> 751 -> 953 (unfortunately it has one 'inactive' digit)

IV. The least carrousel prime with only one digit 'active' is:
821 -> 823 -> 827 -> 829

V. The least carrousel prime with only one 'inactive' digits is: 2111->2333->2777->2999

Questions:

1. Find 3 more carrousel palprimes, having only 'active' digits.

2. Find 3 carrousel primes that generates, each, two couples of reversible primes, having only 'active' digits.

3. Find one Titanic (*) carrousel prime having only one 'active' digit.

4. Find one 100 digits carrousel prime having only one 'inactive' digit.

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(*) As a fair & instructive beginning I can accept smaller solutions if they are composed of 100, 200, ..., etc. digits. As maybe you have noticed these asked four primes may form two couples of twins in the same decade.

Solution

Michael Bell has produced solutions to questions 1 & 3. Here is his email (22/4/01):

"I've found 2 more carrousel palprimes with only active digits:

3193391713171933913 -> 7317713937393177137 -> 9739937179717399379 -> 1971179391939711791 and

7319171773771719137 -> 9731393997993931379 -> 1973717119117173791 -> 3197939331339397913

Also, as a first try at question 3 I found the quadruplet 46060600066600606006*10^30 + 1,3,7,9. Which has 50 digits and only 1 active digit."

Michael is not sure that the palprime gotten is the earliest. He also think that to get a titanic solution to question 3 is "close to impossible, unless someone can see a way of sieving efficiently "

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Giuliano Daddario has solved (2/9/01) the question 2, founding 5 examples with the asked properties:

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