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:

{139397, 371719, 793931, 917173}

{193937, 317179, 739391, 971713}

{1173193939379177,3397317171791399,7719739393913711,9931971717137933}

{113799319379331977, 337911731791773199, 779133973913997311,

991377197137119733}

{139717391739171397, 371939713971393719,

793171937193717931, 917393179317939173}

***

Carlos Rivera wrote on July 13, 2018:

Q1. A Palprime Carrousel Quadruplet, 91 digits all active:
1377373191113371973939373177717339991717199719179917171999337177713739393791733111913737731
3799797313337793197171797399939771113939311931391139393111779399937971717913977333137979973
7911919737779917319393919711171993337171733173713371717333991711179193939137199777379191197
9133131979991139731717131933393117779393977397937793939777113933391317171379311999791313319

***

Carlos Rivera wrote on July 20, 2018:

Q1. A Palprime Carrousel Quadruplet, 97 digits all active:

1997117331197931731939739137111917991979131311333331131319791997191117319379391371397911337117991
3119339773319173973171971379333139113191373733777773373731913119313339731791713793719133779339113
7331771997731397197393193791777371337313797977999997797973137331737771973913937917931377991771337
9773993119973719319717317913999793779737919199111119919197379773979993197137179139173799113993779

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Emmanuel Vantieghem wrote on July 21, 2018:

Your beautiful results about puzzle 134 inspired me to look for palprimes that use only two active digits and that produce a caroussel.

This is my best result (so far) :

Q1. A Palprime Carrousel Quadruplet, 101 digits all active of which only two are used :
11333331313131131331313333131131331333311111131311111313111111333313313113133331313313113131313333311
33777773737373373773737777373373773777733333373733333737333333777737737337377773737737337373737777733
77999997979797797997979999797797997999977777797977777979777777999979979779799997979979779797979999977
99111119191919919119191111919919119111199999919199999191999999111191191991911119191191991919191111199

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